Calculus Examples

$\int \cos^{6} (x) d x$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $2$ out of $6$ .

$\int {\cos (x)}^{2 (3)} d x$

Step 1.2

Rewrite ${\cos (x)}^{2 (3)}$ as exponentiation.

$\int {(\cos^{2} (x))}^{3} d x$

Step 2

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

$\int {(\frac{1 + \cos (2 x)}{2})}^{3} d x$

Step 3

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 $d$ $u_{1}$ .

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Step 3.1

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

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Step 3.1.1

Differentiate $2 x$ .

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

Step 3.1.2

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 3.1.3

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 3.1.4

Multiply $2$ by $1$ .

$2$

Step 3.2

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

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

Step 4

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

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

Step 5

Simplify by multiplying through.

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Step 5.1

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

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

Step 5.2

Expand ${(\frac{1}{2} \cdot (1 + \cos (u_{1})))}^{3}$ .

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Step 5.2.1

Rewrite the exponentiation as a product.

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

Step 5.2.2

Rewrite the exponentiation as a product.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Apply the distributive property.

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

Step 5.2.6

Apply the distributive property.

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

Step 5.2.7

Apply the distributive property.

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

Step 5.2.8

Apply the distributive property.

Step 5.2.9

Apply the distributive property.

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

Step 5.2.10

Apply the distributive property.

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

Step 5.2.11

Apply the distributive property.

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

Step 5.2.12

Apply the distributive property.

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

Step 5.2.13

Apply the distributive property.

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

Step 5.2.14

Apply the distributive property.

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

Step 5.2.15

Apply the distributive property.

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

Step 5.2.16

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

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

Step 5.2.17

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

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

Step 5.2.18

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

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

Step 5.2.19

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

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

Step 5.2.20

Move parentheses.

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

Step 5.2.21

Move parentheses.

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

Step 5.2.22

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

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

Step 5.2.23

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

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

Step 5.2.24

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

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

Step 5.2.25

Move parentheses.

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

Step 5.2.26

Move parentheses.

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

Step 5.2.27

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

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

Step 5.2.28

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

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

Step 5.2.29

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

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

Step 5.2.30

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

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

Step 5.2.31

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

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

Step 5.2.32

Move parentheses.

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

Step 5.2.33

Move parentheses.

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

Step 5.2.34

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

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

Step 5.2.35

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

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

Step 5.2.36

Move parentheses.

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

Step 5.2.37

Move parentheses.

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

Step 5.2.38

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

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

Step 5.2.39

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

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

Step 5.2.40

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

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

Step 5.2.41

Move parentheses.

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

Step 5.2.42

Move parentheses.

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

Step 5.2.43

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

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

Step 5.2.44

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

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

Step 5.2.45

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

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

Step 5.2.46

Move parentheses.

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

Step 5.2.47

Move parentheses.

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

Step 5.2.48

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

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

Step 5.2.49

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

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

Step 5.2.50

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

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

Step 5.2.51

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

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

Step 5.2.52

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

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

Step 5.2.53

Move parentheses.

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

Step 5.2.54

Move parentheses.

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

Step 5.2.55

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

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

Step 5.2.56

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

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

Step 5.2.57

Move parentheses.

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

Step 5.2.58

Move parentheses.

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

Step 5.2.59

Multiply $1$ by $1$ .

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

Step 5.2.60

Multiply $1$ by $1$ .

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

Step 5.2.61

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

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

Step 5.2.62

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

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

Step 5.2.63

Multiply $2$ by $2$ .

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

Step 5.2.64

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

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

Step 5.2.65

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.66

Multiply $1$ by $1$ .

$\frac{1}{2} \int \frac{1}{8} + 1 (\frac{1}{2}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.67

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

$\frac{1}{2} \int \frac{1}{8} + \frac{1}{2} \cdot \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.68

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

$\frac{1}{2} \int \frac{1}{8} + \frac{1}{2 \cdot 2} \cdot \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.69

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{1}{4} \cdot \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.70

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{1}{4 \cdot 2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.71

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{1}{8} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.72

Combine $\frac{1}{8}$ and $\cos (u_{1})$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.73

Multiply $1$ by $1$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.74

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.75

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{2 \cdot 2} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.76

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{4} \cos (u_{1}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.77

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{4} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.78

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{4 \cdot 2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.79

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.80

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.81

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{2 \cdot 2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.82

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{4} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.83

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{4} \cdot \frac{1}{2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.84

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{4 \cdot 2} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.85

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} \cos (u_{1}) + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.86

Combine $\frac{\cos (u_{1})}{8}$ and $\cos (u_{1})$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1}) \cos (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.87

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.88

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.89

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

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{{\cos (u_{1})}^{1 + 1}}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.90

Add $1$ and $1$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.91

Add $\frac{\cos (u_{1})}{8}$ and $\frac{\cos (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{1}{8} + 2 \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.92

Combine $2$ and $\frac{\cos (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.93

Multiply $1$ by $1$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.94

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.95

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.96

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{2 \cdot 2} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.97

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{4} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.98

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{4 \cdot 2} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.99

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.100

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.101

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.102

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{2 \cdot 2} \cdot \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.103

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{4} \cdot \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.104

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{4 \cdot 2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.105

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.106

Combine $\frac{\cos (u_{1})}{8}$ and $\cos (u_{1})$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos (u_{1}) \cos (u_{1})}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.107

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.108

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.109

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{{\cos (u_{1})}^{1 + 1}}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.110

Add $1$ and $1$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.111

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.112

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.113

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{2 \cdot 2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.114

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{4} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.115

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.116

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.117

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.118

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{{\cos (u_{1})}^{1 + 1}}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.119

Add $1$ and $1$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{4} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.120

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{4 \cdot 2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.121

Multiply $4$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.122

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.123

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{2 \cdot 2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.124

Multiply $2$ by $2$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{4} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.125

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.126

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{4} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.127

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{4} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.128

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

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{{\cos (u_{1})}^{1 + 1}}{4} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.129

Add $1$ and $1$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{4} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 5.2.130

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

Step 5.2.131

Multiply $4$ by $2$ .

Step 5.2.132

Combine $\frac{\cos^{2} (u_{1})}{8}$ and $\cos (u_{1})$ .

Step 5.2.133

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

Step 5.2.134

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

Step 5.2.135

Add $2$ and $1$ .

Step 5.2.136

Add $\frac{\cos^{2} (u_{1})}{8}$ and $\frac{\cos^{2} (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + 2 \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} d u_{1}$

Step 5.2.137

Combine $2$ and $\frac{\cos^{2} (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} d u_{1}$

Step 5.2.138

Reorder $\frac{2 \cos (u_{1})}{8}$ and $\frac{\cos^{2} (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{1}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} d u_{1}$

Step 5.2.139

Reorder $\frac{1}{8}$ and $\frac{\cos^{2} (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{\cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} d u_{1}$

Step 5.2.140

Reorder $\frac{2 \cos^{2} (u_{1})}{8}$ and $\frac{\cos^{3} (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{\cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} d u_{1}$

Step 5.2.141

Move $\frac{\cos (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{\cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{\cos (u_{1})}{8} d u_{1}$

Step 5.2.142

Move $\frac{2 \cos (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{\cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{\cos^{3} (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} d u_{1}$

Step 5.2.143

Move $\frac{1}{8}$ .

$\frac{1}{2} \int \frac{\cos^{2} (u_{1})}{8} + \frac{\cos^{3} (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} d u_{1}$

Step 5.2.144

Reorder $\frac{\cos^{2} (u_{1})}{8}$ and $\frac{\cos^{3} (u_{1})}{8}$ .

$\frac{1}{2} \int \frac{\cos^{3} (u_{1})}{8} + \frac{\cos^{2} (u_{1})}{8} + \frac{2 \cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} d u_{1}$

Step 5.2.145

Combine the numerators over the common denominator.

$\frac{1}{2} \int \frac{\cos^{3} (u_{1})}{8} + \frac{\cos^{2} (u_{1}) + 2 \cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} d u_{1}$

Step 5.2.146

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

$\frac{1}{2} \int \frac{\cos^{3} (u_{1})}{8} + \frac{3 \cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1})}{8} + \frac{\cos (u_{1})}{8} d u_{1}$

Step 5.2.147

Combine the numerators over the common denominator.

$\frac{1}{2} \int \frac{\cos^{3} (u_{1})}{8} + \frac{3 \cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{2 \cos (u_{1}) + \cos (u_{1})}{8} d u_{1}$

Step 5.2.148

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

$\frac{1}{2} \int \frac{\cos^{3} (u_{1})}{8} + \frac{3 \cos^{2} (u_{1})}{8} + \frac{1}{8} + \frac{3 \cos (u_{1})}{8} d u_{1}$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{2} (\int \frac{\cos^{3} (u_{1})}{8} d u_{1} + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 7

Since $\frac{1}{8}$ is constant with respect to $u_{1}$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{2} (\frac{1}{8} \int \cos^{3} (u_{1}) d u_{1} + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 8

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

$\frac{1}{2} (\frac{1}{8} \int \cos^{2} (u_{1}) \cos (u_{1}) d u_{1} + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 9

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

$\frac{1}{2} (\frac{1}{8} \int (1 - \sin^{2} (u_{1})) \cos (u_{1}) d u_{1} + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

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 $d$ $u_{2}$ .

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Step 10.1

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

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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 $\cos (u_{1})$ .

$\cos (u_{1})$

Step 10.2

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

$\frac{1}{2} (\frac{1}{8} \int 1 - {u_{2}}^{2} d u_{2} + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 11

Split the single integral into multiple integrals.

$\frac{1}{2} (\frac{1}{8} (\int d u_{2} + \int - {u_{2}}^{2} d u_{2}) + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 12

Apply the constant rule.

$\frac{1}{2} (\frac{1}{8} (u_{2} + C + \int - {u_{2}}^{2} d u_{2}) + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 13

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - \int {u_{2}}^{2} d u_{2}) + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 14

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{1}{3} {u_{2}}^{3} + C)) + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 15

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \int \frac{3 \cos^{2} (u_{1})}{8} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 16

Since $\frac{3}{8}$ is constant with respect to $u_{1}$ , move $\frac{3}{8}$ out of the integral.

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{8} \int \cos^{2} (u_{1}) d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 17

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{8} \int \frac{1 + \cos (2 u_{1})}{2} d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 18

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{8} (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1}) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 19

Simplify.

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Step 19.1

Multiply $\frac{1}{2}$ by $\frac{3}{8}$ .

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{2 \cdot 8} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 19.2

Multiply $2$ by $8$ .

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 20

Split the single integral into multiple integrals.

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1}) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 21

Apply the constant rule.

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \int \cos (2 u_{1}) d u_{1}) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 22

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 $d$ $u_{3}$ .

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Step 22.1

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

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Step 22.1.1

Differentiate $2 u_{1}$ .

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

Step 22.1.2

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 22.1.3

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 22.1.4

Multiply $2$ by $1$ .

$2$

Step 22.2

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \int \cos (u_{3}) \frac{1}{2} d u_{3}) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 23

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \int \frac{\cos (u_{3})}{2} d u_{3}) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 24

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \frac{1}{2} \int \cos (u_{3}) d u_{3}) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 25

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \int \frac{1}{8} d u_{1} + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 26

Apply the constant rule.

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{1}{8} u_{1} + C + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 27

Combine $\frac{1}{8}$ and $u_{1}$ .

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{u_{1}}{8} + C + \int \frac{3 \cos (u_{1})}{8} d u_{1})$

Step 28

Since $\frac{3}{8}$ is constant with respect to $u_{1}$ , move $\frac{3}{8}$ out of the integral.

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{u_{1}}{8} + C + \frac{3}{8} \int \cos (u_{1}) d u_{1})$

Step 29

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

$\frac{1}{2} (\frac{1}{8} (u_{2} + C - (\frac{{u_{2}}^{3}}{3} + C)) + \frac{3}{16} (u_{1} + C + \frac{1}{2} (\sin (u_{3}) + C)) + \frac{u_{1}}{8} + C + \frac{3}{8} (\sin (u_{1}) + C))$

Step 30

Simplify.

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Step 30.1

Simplify.

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{3 u_{1}}{16} + \frac{3 \sin (u_{3})}{32} + \frac{u_{1}}{8} + \frac{3 \sin (u_{1})}{8}) + C$

Step 30.2

Simplify.

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Step 30.2.1

To write $\frac{u_{1}}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{3 u_{1}}{16} + \frac{u_{1}}{8} \cdot \frac{2}{2} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 30.2.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 30.2.2.1

Multiply $\frac{u_{1}}{8}$ by $\frac{2}{2}$ .

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{3 u_{1}}{16} + \frac{u_{1} \cdot 2}{8 \cdot 2} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 30.2.2.2

Multiply $8$ by $2$ .

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{3 u_{1}}{16} + \frac{u_{1} \cdot 2}{16} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 30.2.3

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{3 u_{1} + u_{1} \cdot 2}{16} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 30.2.4

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

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{3 u_{1} + 2 \cdot u_{1}}{16} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 30.2.5

Add $3 u_{1}$ and $2 u_{1}$ .

$\frac{1}{2} (\frac{u_{2}}{8} - \frac{{u_{2}}^{3}}{24} + \frac{5 u_{1}}{16} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 31

Substitute back in for each integration substitution variable.

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Step 31.1

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

$\frac{1}{2} (\frac{\sin (u_{1})}{8} - \frac{\sin^{3} (u_{1})}{24} + \frac{5 u_{1}}{16} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (u_{1})}{8}) + C$

Step 31.2

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

$\frac{1}{2} (\frac{\sin (2 x)}{8} - \frac{\sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (u_{3})}{32} + \frac{3 \sin (2 x)}{8}) + C$

Step 31.3

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

$\frac{1}{2} (\frac{\sin (2 x)}{8} - \frac{\sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 u_{1})}{32} + \frac{3 \sin (2 x)}{8}) + C$

Step 31.4

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

$\frac{1}{2} (\frac{\sin (2 x)}{8} - \frac{\sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32} + \frac{3 \sin (2 x)}{8}) + C$

Step 32

Simplify.

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Step 32.1

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{\sin (2 x) + 3 \sin (2 x)}{8} + \frac{- \sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.2

Add $\sin (2 x)$ and $3 \sin (2 x)$ .

$\frac{1}{2} (\frac{4 \sin (2 x)}{8} + \frac{- \sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3

Simplify each term.

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Step 32.3.1

Cancel the common factor of $4$ and $8$ .

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Step 32.3.1.1

Factor $4$ out of $4 \sin (2 x)$ .

$\frac{1}{2} (\frac{4 (\sin (2 x))}{8} + \frac{- \sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.1.2

Cancel the common factors.

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Step 32.3.1.2.1

Factor $4$ out of $8$ .

$\frac{1}{2} (\frac{4 \sin (2 x)}{4 \cdot 2} + \frac{- \sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.1.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{4 \sin (2 x)}{4 \cdot 2} + \frac{- \sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.1.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{\sin (2 x)}{2} + \frac{- \sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.2

Move the negative in front of the fraction.

$\frac{1}{2} (\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{24} + \frac{5 (2 x)}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.3

Cancel the common factor of $2$ and $16$ .

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Step 32.3.3.1

Factor $2$ out of $5 (2 x)$ .

$\frac{1}{2} (\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{24} + \frac{2 (5 (x))}{16} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.3.2

Cancel the common factors.

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Step 32.3.3.2.1

Factor $2$ out of $16$ .

$\frac{1}{2} (\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{24} + \frac{2 (5 (x))}{2 \cdot 8} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.3.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{24} + \frac{2 (5 (x))}{2 \cdot 8} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.3.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{24} + \frac{5 (x)}{8} + \frac{3 \sin (2 (2 x))}{32}) + C$

Step 32.3.4

Multiply $2$ by $2$ .

$\frac{1}{2} (\frac{\sin (2 x)}{2} - \frac{\sin^{3} (2 x)}{24} + \frac{5 x}{8} + \frac{3 \sin (4 x)}{32}) + C$

Step 32.4

Apply the distributive property.

$\frac{1}{2} \cdot \frac{\sin (2 x)}{2} + \frac{1}{2} (- \frac{\sin^{3} (2 x)}{24}) + \frac{1}{2} \cdot \frac{5 x}{8} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5

Simplify.

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Step 32.5.1

Multiply $\frac{1}{2} \cdot \frac{\sin (2 x)}{2}$ .

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Step 32.5.1.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 x)}{2}$ .

$\frac{\sin (2 x)}{2 \cdot 2} + \frac{1}{2} (- \frac{\sin^{3} (2 x)}{24}) + \frac{1}{2} \cdot \frac{5 x}{8} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5.1.2

Multiply $2$ by $2$ .

$\frac{\sin (2 x)}{4} + \frac{1}{2} (- \frac{\sin^{3} (2 x)}{24}) + \frac{1}{2} \cdot \frac{5 x}{8} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5.2

Multiply $\frac{1}{2} (- \frac{\sin^{3} (2 x)}{24})$ .

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Step 32.5.2.1

Multiply $\frac{1}{2}$ by $\frac{\sin^{3} (2 x)}{24}$ .

$\frac{\sin (2 x)}{4} - \frac{\sin^{3} (2 x)}{2 \cdot 24} + \frac{1}{2} \cdot \frac{5 x}{8} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5.2.2

Multiply $2$ by $24$ .

$\frac{\sin (2 x)}{4} - \frac{\sin^{3} (2 x)}{48} + \frac{1}{2} \cdot \frac{5 x}{8} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5.3

Multiply $\frac{1}{2} \cdot \frac{5 x}{8}$ .

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Step 32.5.3.1

Multiply $\frac{1}{2}$ by $\frac{5 x}{8}$ .

$\frac{\sin (2 x)}{4} - \frac{\sin^{3} (2 x)}{48} + \frac{5 x}{2 \cdot 8} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5.3.2

Multiply $2$ by $8$ .

$\frac{\sin (2 x)}{4} - \frac{\sin^{3} (2 x)}{48} + \frac{5 x}{16} + \frac{1}{2} \cdot \frac{3 \sin (4 x)}{32} + C$

Step 32.5.4

Multiply $\frac{1}{2} \cdot \frac{3 \sin (4 x)}{32}$ .

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Step 32.5.4.1

Multiply $\frac{1}{2}$ by $\frac{3 \sin (4 x)}{32}$ .

$\frac{\sin (2 x)}{4} - \frac{\sin^{3} (2 x)}{48} + \frac{5 x}{16} + \frac{3 \sin (4 x)}{2 \cdot 32} + C$

Step 32.5.4.2

Multiply $2$ by $32$ .

$\frac{\sin (2 x)}{4} - \frac{\sin^{3} (2 x)}{48} + \frac{5 x}{16} + \frac{3 \sin (4 x)}{64} + C$

Step 33

Reorder terms.

$\frac{1}{4} \sin (2 x) - \frac{1}{48} \sin^{3} (2 x) + \frac{5}{16} x + \frac{3}{64} \sin (4 x) + C$