Calculus Examples

$\int \sin^{4} (x) \cos^{2} (x) d x$

Step 1

Use the half-angle formula to rewrite $\cos^{2} (x)$ as $\frac{1 + \cos (2 x)}{2}$ .

$\int \sin^{4} (x) \frac{1 + \cos (2 x)}{2} d x$

Step 2

Simplify with factoring out.

Tap for more steps...

Step 2.1

Factor $2$ out of $4$ .

$\int {\sin (x)}^{2 (2)} \frac{1 + \cos (2 x)}{2} d x$

Step 2.2

Rewrite ${\sin (x)}^{2 (2)}$ as exponentiation.

$\int {(\sin^{2} (x))}^{2} \frac{1 + \cos (2 x)}{2} d x$

$\int {(\sin^{2} (x))}^{2} \frac{1 + \cos (2 x)}{2} d x$

Step 3

Use the half-angle formula to rewrite $\sin^{2} (x)$ as $\frac{1 - \cos (2 x)}{2}$ .

$\int {(\frac{1 - \cos (2 x)}{2})}^{2} \frac{1 + \cos (2 x)}{2} d x$

Step 4

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 4.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 4.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

$2$

Step 4.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1 + \cos (u_{1})}{2} \frac{1}{2} d u_{1}$

$\int {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1 + \cos (u_{1})}{2} \frac{1}{2} d u_{1}$

Step 5

Simplify by multiplying through.

Tap for more steps...

Step 5.1

Simplify.

Tap for more steps...

Step 5.1.1

Multiply $\frac{1}{2}$ by $\frac{1 + \cos (u_{1})}{2}$ .

$\int {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1 + \cos (u_{1})}{2 \cdot 2} d u_{1}$

Step 5.1.2

Multiply $2$ by $2$ .

$\int {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1 + \cos (u_{1})}{4} d u_{1}$

$\int {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1 + \cos (u_{1})}{4} d u_{1}$

Step 5.2

Simplify with commuting.

Tap for more steps...

Step 5.2.1

Rewrite $\frac{1 - \cos (u_{1})}{2}$ as a product.

$\int {(\frac{1}{2} \cdot (1 - \cos (u_{1})))}^{2} \frac{1 + \cos (u_{1})}{4} d u_{1}$

Step 5.2.2

Rewrite $\frac{1 + \cos (u_{1})}{4}$ as a product.

$\int {(\frac{1}{2} \cdot (1 - \cos (u_{1})))}^{2} (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

$\int {(\frac{1}{2} \cdot (1 - \cos (u_{1})))}^{2} (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3

Expand ${(\frac{1}{2} \cdot (1 - \cos (u_{1})))}^{2} (\frac{1}{4} \cdot (1 + \cos (u_{1})))$ .

Tap for more steps...

Step 5.3.1

Rewrite the exponentiation as a product.

$\int \frac{1}{2} \cdot (1 - \cos (u_{1})) (\frac{1}{2} \cdot (1 - \cos (u_{1}))) (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3.2

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{2} \cdot (1 - \cos (u_{1}))) (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3.3

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3.4

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3.5

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3.6

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 5.3.7

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.8

Apply the distributive property.

$\int (\frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) + (\frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.9

Apply the distributive property.

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) + (\frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.10

Apply the distributive property.

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) + (\frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.11

Apply the distributive property.

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + (\frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1})))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.12

Apply the distributive property.

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.13

Apply the distributive property.

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1 + \frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.14

Apply the distributive property.

$\int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.15

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.16

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.17

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.18

Reorder $\frac{1}{4}$ and $1$ .

$\int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} (1 \cdot \frac{1}{4}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.19

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \frac{1}{2} \cdot 1 \frac{1}{2} \cdot \frac{1}{4} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.20

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.21

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.22

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.23

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.24

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.25

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2} \cos (u_{1})) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.26

Move parentheses.

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2}) \cos (u_{1}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.27

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.28

Reorder $\frac{1}{4}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) (1 \cdot \frac{1}{4}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.29

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cdot 1 \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.30

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \frac{1}{2} \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.31

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.32

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.33

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2} \cos (u_{1})) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.34

Move parentheses.

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2}) \cos (u_{1}) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.35

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.36

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.37

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (1 \cdot \frac{1}{2}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.38

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cdot 1 \cos (u_{1}) \cdot \frac{1}{2} (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.39

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.40

Reorder $\frac{1}{4}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} (1 \cdot \frac{1}{4}) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.41

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot 1 \frac{1}{2} \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.42

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cdot 1 \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.43

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.44

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{2} \cdot 1) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.45

Reorder $\frac{1}{2}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cos (u_{1}) (1 \cdot \frac{1}{2}) (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.46

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cdot 1 \cos (u_{1}) \cdot \frac{1}{2} (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.47

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} (\frac{1}{4} \cdot \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.48

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.49

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (- 1 \cdot \frac{1}{2} \cos (u_{1})) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.50

Move parentheses.

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (- 1 \cdot \frac{1}{2}) \cos (u_{1}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.51

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot \frac{1}{2} \cdot - 1 \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.52

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{4} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.53

Reorder $\frac{1}{4}$ and $1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) (1 \cdot \frac{1}{4}) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.54

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \cdot 1 \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.55

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.56

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \frac{1}{2} \cdot 1 \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.57

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.58

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.59

Reorder $\frac{1}{2}$ and $- 1$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cos (u_{1}) (- 1 \cdot \frac{1}{2} \cos (u_{1})) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.60

Move parentheses.

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cos (u_{1}) (- 1 \cdot \frac{1}{2}) \cos (u_{1}) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.61

Move $\cos (u_{1})$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} - 1 \cdot \frac{1}{2} \cdot - 1 \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.62

Move $\frac{1}{2}$ .

$\int 1 \cdot 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \frac{1}{4} \cdot \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{4} \cdot \cos (u_{1})) d u_{1}$

Step 5.3.63

Multiply $1$ by $1$ .

$\int 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.64

Multiply $1$ by $1$ .

$\int 1 (\frac{1}{2}) \frac{1}{2} \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.65

Multiply $\frac{1}{2}$ by $1$ .

$\int \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.66

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\int \frac{1}{2 \cdot 2} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.67

Multiply $2$ by $2$ .

$\int \frac{1}{4} \cdot \frac{1}{4} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.68

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$\int \frac{1}{4 \cdot 4} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.69

Multiply $4$ by $4$ .

$\int \frac{1}{16} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.70

Multiply $1$ by $1$ .

$\int \frac{1}{16} + 1 (\frac{1}{2}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.71

Multiply $\frac{1}{2}$ by $1$ .

$\int \frac{1}{16} + \frac{1}{2} \cdot \frac{1}{2} \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.72

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\int \frac{1}{16} + \frac{1}{2 \cdot 2} \cdot \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.73

Multiply $2$ by $2$ .

$\int \frac{1}{16} + \frac{1}{4} \cdot \frac{1}{4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.74

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

$\int \frac{1}{16} + \frac{1}{4 \cdot 4} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.75

Multiply $4$ by $4$ .

$\int \frac{1}{16} + \frac{1}{16} \cos (u_{1}) + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.76

Combine $\frac{1}{16}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} + 1 \cdot - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.77

Multiply $- 1$ by $1$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.78

Multiply $- 1$ by $1$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.79

Combine $- \frac{1}{2}$ and $\frac{1}{2}$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{1}{2 \cdot 2} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.80

Multiply $2$ by $2$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{1}{4} \cos (u_{1}) \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.81

Combine $\frac{1}{4}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{4} \cdot \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.82

Combine $- \frac{\cos (u_{1})}{4}$ and $\frac{1}{4}$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{4 \cdot 4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.83

Multiply $4$ by $4$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.84

Multiply $- 1$ by $1$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.85

Combine $- \frac{1}{2}$ and $\frac{1}{2}$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{1}{2 \cdot 2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.86

Multiply $2$ by $2$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{1}{4} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.87

Combine $\frac{1}{4}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{4} \cdot \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.88

Combine $- \frac{\cos (u_{1})}{4}$ and $\frac{1}{4}$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{4 \cdot 4} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.89

Multiply $4$ by $4$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} \cos (u_{1}) - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.90

Combine $\frac{\cos (u_{1})}{16}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1}) \cos (u_{1})}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.91

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{1} (u_{1}) \cos (u_{1})}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.92

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.93

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{{\cos (u_{1})}^{1 + 1}}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.94

Add $1$ and $1$ .

$\int \frac{1}{16} + \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.95

Subtract $\frac{\cos (u_{1})}{16}$ from $\frac{\cos (u_{1})}{16}$ .

$\int \frac{1}{16} + 0 - \frac{\cos^{2} (u_{1})}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.96

Subtract $\frac{\cos^{2} (u_{1})}{16}$ from $0$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - 1 \cdot 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.97

Multiply $- 1$ by $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.98

Multiply $- 1$ by $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.99

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.100

Combine $- \frac{\cos (u_{1})}{2}$ and $\frac{1}{2}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{2 \cdot 2} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.101

Multiply $2$ by $2$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{4} \cdot \frac{1}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.102

Combine $- \frac{\cos (u_{1})}{4}$ and $\frac{1}{4}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{4 \cdot 4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.103

Multiply $4$ by $4$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.104

Multiply $- 1$ by $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.105

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.106

Combine $- \frac{\cos (u_{1})}{2}$ and $\frac{1}{2}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{2 \cdot 2} \cdot \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.107

Multiply $2$ by $2$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{4} \cdot \frac{1}{4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.108

Combine $- \frac{\cos (u_{1})}{4}$ and $\frac{1}{4}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{4 \cdot 4} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.109

Multiply $4$ by $4$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1})}{16} \cos (u_{1}) - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.110

Combine $\frac{\cos (u_{1})}{16}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos (u_{1}) \cos (u_{1})}{16} - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.111

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{1} (u_{1}) \cos (u_{1})}{16} - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.112

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{16} - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.113

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{{\cos (u_{1})}^{1 + 1}}{16} - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.114

Add $1$ and $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} - 1 \cdot - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.115

Multiply $- 1$ by $- 1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.116

Multiply $1$ by $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.117

Multiply $\frac{1}{2}$ by $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.118

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.119

Multiply $\frac{\cos (u_{1})}{2}$ by $\frac{1}{2}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1})}{2 \cdot 2} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.120

Multiply $2$ by $2$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1})}{4} \cos (u_{1}) \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.121

Combine $\frac{\cos (u_{1})}{4}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.122

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.123

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{4} \cdot \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.124

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{{\cos (u_{1})}^{1 + 1}}{4} \cdot \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.125

Add $1$ and $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{4} \cdot \frac{1}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.126

Multiply $\frac{\cos^{2} (u_{1})}{4}$ by $\frac{1}{4}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{4 \cdot 4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.127

Multiply $4$ by $4$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.128

Multiply $- 1$ by $- 1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.129

Multiply $\frac{1}{2}$ by $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.130

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.131

Multiply $\frac{\cos (u_{1})}{2}$ by $\frac{1}{2}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1})}{2 \cdot 2} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.132

Multiply $2$ by $2$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1})}{4} \cos (u_{1}) \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.133

Combine $\frac{\cos (u_{1})}{4}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.134

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.135

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{4} \cdot \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.136

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{{\cos (u_{1})}^{1 + 1}}{4} \cdot \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.137

Add $1$ and $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{4} \cdot \frac{1}{4} \cos (u_{1}) d u_{1}$

Step 5.3.138

Multiply $\frac{\cos^{2} (u_{1})}{4}$ by $\frac{1}{4}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{4 \cdot 4} \cos (u_{1}) d u_{1}$

Step 5.3.139

Multiply $4$ by $4$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} \cos (u_{1}) d u_{1}$

Step 5.3.140

Combine $\frac{\cos^{2} (u_{1})}{16}$ and $\cos (u_{1})$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1}) \cos (u_{1})}{16} d u_{1}$

Step 5.3.141

Raise $\cos (u_{1})$ to the power of $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1}) \cos^{1} (u_{1})}{16} d u_{1}$

Step 5.3.142

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{{\cos (u_{1})}^{2 + 1}}{16} d u_{1}$

Step 5.3.143

Add $2$ and $1$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{3} (u_{1})}{16} d u_{1}$

Step 5.3.144

Add $- \frac{\cos^{2} (u_{1})}{16}$ and $\frac{\cos^{2} (u_{1})}{16}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} + 0 + \frac{\cos^{3} (u_{1})}{16} d u_{1}$

Step 5.3.145

Add $0$ and $\frac{\cos^{3} (u_{1})}{16}$ .

$\int \frac{1}{16} - \frac{\cos^{2} (u_{1})}{16} - \frac{\cos (u_{1})}{16} + \frac{\cos^{3} (u_{1})}{16} d u_{1}$

Step 5.3.146

Reorder $\frac{1}{16}$ and $- \frac{\cos^{2} (u_{1})}{16}$ .

$\int - \frac{\cos^{2} (u_{1})}{16} + \frac{1}{16} - \frac{\cos (u_{1})}{16} + \frac{\cos^{3} (u_{1})}{16} d u_{1}$

Step 5.3.147

Reorder $- \frac{\cos (u_{1})}{16}$ and $\frac{\cos^{3} (u_{1})}{16}$ .

$\int - \frac{\cos^{2} (u_{1})}{16} + \frac{1}{16} + \frac{\cos^{3} (u_{1})}{16} - \frac{\cos (u_{1})}{16} d u_{1}$

Step 5.3.148

Move $\frac{1}{16}$ .

$\int - \frac{\cos^{2} (u_{1})}{16} + \frac{\cos^{3} (u_{1})}{16} + \frac{1}{16} - \frac{\cos (u_{1})}{16} d u_{1}$

Step 5.3.149

Reorder $- \frac{\cos^{2} (u_{1})}{16}$ and $\frac{\cos^{3} (u_{1})}{16}$ .

$\int \frac{\cos^{3} (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{1}{16} - \frac{\cos (u_{1})}{16} d u_{1}$

$\int \frac{\cos^{3} (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{1}{16} - \frac{\cos (u_{1})}{16} d u_{1}$

$\int \frac{\cos^{3} (u_{1})}{16} - \frac{\cos^{2} (u_{1})}{16} + \frac{1}{16} - \frac{\cos (u_{1})}{16} d u_{1}$

Step 6

Split the single integral into multiple integrals.

$\int \frac{\cos^{3} (u_{1})}{16} d u_{1} + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 7

Since $\frac{1}{16}$ is constant with respect to $u_{1}$ , move $\frac{1}{16}$ out of the integral.

$\frac{1}{16} \int \cos^{3} (u_{1}) d u_{1} + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 8

Factor out $\cos^{2} (u_{1})$ .

$\frac{1}{16} \int \cos^{2} (u_{1}) \cos (u_{1}) d u_{1} + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 9

Using the Pythagorean Identity, rewrite $\cos^{2} (u_{1})$ as $1 - \sin^{2} (u_{1})$ .

$\frac{1}{16} \int (1 - \sin^{2} (u_{1})) \cos (u_{1}) d u_{1} + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 10

Let $u_{2} = \sin (u_{1})$ . Then $d u_{2} = \cos (u_{1}) d u_{1}$ , so $\frac{1}{\cos (u_{1})} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 10.1

Let $u_{2} = \sin (u_{1})$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 10.1.1

Differentiate $\sin (u_{1})$ .

$\frac{d}{d u_{1}} [\sin (u_{1})]$

Step 10.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1})$

$\cos (u_{1})$

Step 10.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{16} \int 1 - {u_{2}}^{2} d u_{2} + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

$\frac{1}{16} \int 1 - {u_{2}}^{2} d u_{2} + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 11

Split the single integral into multiple integrals.

$\frac{1}{16} (\int d u_{2} + \int - {u_{2}}^{2} d u_{2}) + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 12

Apply the constant rule.

$\frac{1}{16} (u_{2} + C + \int - {u_{2}}^{2} d u_{2}) + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 13

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$\frac{1}{16} (u_{2} + C - \int {u_{2}}^{2} d u_{2}) + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 14

By the Power Rule, the integral of ${u_{2}}^{2}$ with respect to $u_{2}$ is $\frac{1}{3} {u_{2}}^{3}$ .

$\frac{1}{16} (u_{2} + C - (\frac{1}{3} {u_{2}}^{3} + C)) + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 15

Combine $\frac{1}{3}$ and ${u_{2}}^{3}$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \int - \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 16

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \int \frac{\cos^{2} (u_{1})}{16} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 17

Since $\frac{1}{16}$ is constant with respect to $u_{1}$ , move $\frac{1}{16}$ out of the integral.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - (\frac{1}{16} \int \cos^{2} (u_{1}) d u_{1}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 18

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{16} \int \frac{1 + \cos (2 u_{1})}{2} d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 19

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{16} (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 20

Simplify.

Tap for more steps...

Step 20.1

Multiply $\frac{1}{2}$ by $\frac{1}{16}$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{2 \cdot 16} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 20.2

Multiply $2$ by $16$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 21

Split the single integral into multiple integrals.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 22

Apply the constant rule.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \int \cos (2 u_{1}) d u_{1}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 23

Let $u_{3} = 2 u_{1}$ . Then $d u_{3} = 2 d u_{1}$ , so $\frac{1}{2} d u_{3} = d u_{1}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

Tap for more steps...

Step 23.1

Let $u_{3} = 2 u_{1}$ . Find $\frac{d u_{3}}{d u_{1}}$ .

Tap for more steps...

Step 23.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 23.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 23.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 23.1.4

Multiply $2$ by $1$ .

$2$

$2$

Step 23.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \int \cos (u_{3}) \frac{1}{2} d u_{3}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \int \cos (u_{3}) \frac{1}{2} d u_{3}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 24

Combine $\cos (u_{3})$ and $\frac{1}{2}$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \int \frac{\cos (u_{3})}{2} d u_{3}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 25

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \frac{1}{2} \int \cos (u_{3}) d u_{3}) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 26

The integral of $\cos (u_{3})$ with respect to $u_{3}$ is $\sin (u_{3})$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \int \frac{1}{16} d u_{1} + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 27

Apply the constant rule.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{1}{16} u_{1} + C + \int - \frac{\cos (u_{1})}{16} d u_{1}$

Step 28

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{1}{16} u_{1} + C - \int \frac{\cos (u_{1})}{16} d u_{1}$

Step 29

Since $\frac{1}{16}$ is constant with respect to $u_{1}$ , move $\frac{1}{16}$ out of the integral.

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{1}{16} u_{1} + C - (\frac{1}{16} \int \cos (u_{1}) d u_{1})$

Step 30

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$\frac{1}{16} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) - \frac{1}{32} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{1}{16} u_{1} + C - \frac{1}{16} (\sin (u_{1}) + C)$

Step 31

Simplify.

Tap for more steps...

Step 31.1

Simplify.

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} - \frac{u_{1}}{32} - \frac{\sin (u_{3})}{64} + \frac{u_{1}}{16} - \frac{1}{16} \sin (u_{1}) + C$

Step 31.2

Simplify.

Tap for more steps...

Step 31.2.1

To write $\frac{u_{1}}{16}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} - \frac{u_{1}}{32} + \frac{u_{1}}{16} \cdot \frac{2}{2} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 31.2.2

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 31.2.2.1

Multiply $\frac{u_{1}}{16}$ by $\frac{2}{2}$ .

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} - \frac{u_{1}}{32} + \frac{u_{1} \cdot 2}{16 \cdot 2} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 31.2.2.2

Multiply $16$ by $2$ .

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} - \frac{u_{1}}{32} + \frac{u_{1} \cdot 2}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} - \frac{u_{1}}{32} + \frac{u_{1} \cdot 2}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 31.2.3

Combine the numerators over the common denominator.

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} + \frac{- u_{1} + u_{1} \cdot 2}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 31.2.4

Move $2$ to the left of $u_{1}$ .

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} + \frac{- u_{1} + 2 \cdot u_{1}}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 31.2.5

Add $- u_{1}$ and $2 u_{1}$ .

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} + \frac{u_{1}}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} + \frac{u_{1}}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

$\frac{u_{2}}{16} - \frac{{u_{2}}^{3}}{48} + \frac{u_{1}}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 32

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 32.1

Replace all occurrences of $u_{2}$ with $\sin (u_{1})$ .

$\frac{\sin (u_{1})}{16} - \frac{\sin^{3} (u_{1})}{48} + \frac{u_{1}}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (u_{1}) + C$

Step 32.2

Replace all occurrences of $u_{1}$ with $2 x$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 x}{32} - \frac{\sin (u_{3})}{64} - \frac{1}{16} \sin (2 x) + C$

Step 32.3

Replace all occurrences of $u_{3}$ with $2 u_{1}$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 x}{32} - \frac{\sin (2 u_{1})}{64} - \frac{1}{16} \sin (2 x) + C$

Step 32.4

Replace all occurrences of $u_{1}$ with $2 x$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 x}{32} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 x}{32} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

Step 33

Simplify.

Tap for more steps...

Step 33.1

Reduce the expression $\frac{2 x}{32}$ by cancelling the common factors.

Tap for more steps...

Step 33.1.1

Factor $2$ out of $2 x$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 (x)}{32} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

Step 33.1.2

Factor $2$ out of $32$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 x}{2 \cdot 16} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

Step 33.1.3

Cancel the common factor.

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{2 x}{2 \cdot 16} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

Step 33.1.4

Rewrite the expression.

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{x}{16} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{x}{16} - \frac{\sin (2 (2 x))}{64} - \frac{1}{16} \sin (2 x) + C$

Step 33.2

Multiply $2$ by $2$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{x}{16} - \frac{\sin (4 x)}{64} - \frac{1}{16} \sin (2 x) + C$

Step 33.3

Combine $\sin (2 x)$ and $\frac{1}{16}$ .

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{x}{16} - \frac{\sin (4 x)}{64} - \frac{\sin (2 x)}{16} + C$

$\frac{\sin (2 x)}{16} - \frac{\sin^{3} (2 x)}{48} + \frac{x}{16} - \frac{\sin (4 x)}{64} - \frac{\sin (2 x)}{16} + C$

Step 34

Reorder terms.

$\frac{1}{16} \sin (2 x) - \frac{1}{48} \sin^{3} (2 x) + \frac{1}{16} x - \frac{1}{64} \sin (4 x) - \frac{1}{16} \sin (2 x) + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$