Calculus Examples

$\int \cos^{6} (\frac{y}{2}) d y$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\frac{y}{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$\frac{1}{2} \cdot 1$

Step 1.1.4

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

$\frac{1}{2}$

Step 1.2

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

$\int \cos^{6} (u_{1}) \frac{1}{\frac{1}{2}} d u_{1}$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int \cos^{6} (u_{1}) (1 \cdot 2) d u_{1}$

Step 2.2

Multiply $2$ by $1$ .

$\int \cos^{6} (u_{1}) \cdot 2 d u_{1}$

Step 2.3

Move $2$ to the left of $\cos^{6} (u_{1})$ .

$\int 2 \cos^{6} (u_{1}) d u_{1}$

Step 3

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

$2 \int \cos^{6} (u_{1}) d u_{1}$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $2$ out of $6$ .

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

Step 4.2

Rewrite ${\cos (u_{1})}^{2 (3)}$ as exponentiation.

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 u_{1}$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify by multiplying through.

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

Cancel the common factor of $2$ .

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Step 8.1.2.1

Cancel the common factor.

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

Step 8.1.2.2

Rewrite the expression.

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

Step 8.1.3

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

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

Rewrite the exponentiation as a product.

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

Step 8.3.2

Rewrite the exponentiation as a product.

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

Step 8.3.3

Apply the distributive property.

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

Step 8.3.4

Apply the distributive property.

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

Step 8.3.5

Apply the distributive property.

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

Step 8.3.6

Apply the distributive property.

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

Step 8.3.7

Apply the distributive property.

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

Step 8.3.8

Apply the distributive property.

Step 8.3.9

Apply the distributive property.

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

Step 8.3.10

Apply the distributive property.

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

Step 8.3.11

Apply the distributive property.

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

Step 8.3.12

Apply the distributive property.

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

Step 8.3.13

Apply the distributive property.

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

Step 8.3.14

Apply the distributive property.

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

Step 8.3.15

Apply the distributive property.

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

Step 8.3.16

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

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

Step 8.3.17

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

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

Step 8.3.18

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

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

Step 8.3.19

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

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

Step 8.3.20

Move parentheses.

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

Step 8.3.21

Move parentheses.

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

Step 8.3.22

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

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

Step 8.3.23

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

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

Step 8.3.24

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

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

Step 8.3.25

Move parentheses.

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

Step 8.3.26

Move parentheses.

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

Step 8.3.27

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

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

Step 8.3.28

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

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

Step 8.3.29

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

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

Step 8.3.30

Move $\cos (u_{2})$ .

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

Step 8.3.31

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

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

Step 8.3.32

Move parentheses.

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

Step 8.3.33

Move parentheses.

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

Step 8.3.34

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

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

Step 8.3.35

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

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

Step 8.3.36

Move parentheses.

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

Step 8.3.37

Move parentheses.

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

Step 8.3.38

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

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

Step 8.3.39

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

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

Step 8.3.40

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

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

Step 8.3.41

Move parentheses.

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

Step 8.3.42

Move parentheses.

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

Step 8.3.43

Move $\cos (u_{2})$ .

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

Step 8.3.44

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

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

Step 8.3.45

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

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

Step 8.3.46

Move parentheses.

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

Step 8.3.47

Move parentheses.

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

Step 8.3.48

Move $\cos (u_{2})$ .

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

Step 8.3.49

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

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

Step 8.3.50

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

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

Step 8.3.51

Move $\cos (u_{2})$ .

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

Step 8.3.52

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

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

Step 8.3.53

Move parentheses.

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

Step 8.3.54

Move parentheses.

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

Step 8.3.55

Move $\cos (u_{2})$ .

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

Step 8.3.56

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

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

Step 8.3.57

Move parentheses.

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

Step 8.3.58

Move parentheses.

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

Step 8.3.59

Multiply $1$ by $1$ .

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

Step 8.3.60

Multiply $1$ by $1$ .

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

Step 8.3.61

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

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

Step 8.3.62

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

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

Step 8.3.63

Multiply $2$ by $2$ .

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

Step 8.3.64

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

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

Step 8.3.65

Multiply $4$ by $2$ .

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

Step 8.3.66

Multiply $1$ by $1$ .

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

Step 8.3.67

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

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

Step 8.3.68

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

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

Step 8.3.69

Multiply $2$ by $2$ .

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

Step 8.3.70

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

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

Step 8.3.71

Multiply $4$ by $2$ .

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

Step 8.3.72

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

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

Step 8.3.73

Multiply $1$ by $1$ .

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

Step 8.3.74

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

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

Step 8.3.75

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

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

Step 8.3.76

Multiply $2$ by $2$ .

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

Step 8.3.77

Combine $\frac{1}{4}$ and $\cos (u_{2})$ .

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

Step 8.3.78

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

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

Step 8.3.79

Multiply $4$ by $2$ .

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

Step 8.3.80

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

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

Step 8.3.81

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

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

Step 8.3.82

Multiply $2$ by $2$ .

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

Step 8.3.83

Combine $\frac{1}{4}$ and $\cos (u_{2})$ .

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

Step 8.3.84

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

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

Step 8.3.85

Multiply $4$ by $2$ .

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

Step 8.3.86

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

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

Step 8.3.87

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.88

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.89

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

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

Step 8.3.90

Add $1$ and $1$ .

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

Step 8.3.91

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

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

Step 8.3.92

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

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

Step 8.3.93

Multiply $1$ by $1$ .

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

Step 8.3.94

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

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

Step 8.3.95

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

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

Step 8.3.96

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

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

Step 8.3.97

Multiply $2$ by $2$ .

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

Step 8.3.98

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

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

Step 8.3.99

Multiply $4$ by $2$ .

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

Step 8.3.100

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

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

Step 8.3.101

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

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

Step 8.3.102

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

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

Step 8.3.103

Multiply $2$ by $2$ .

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

Step 8.3.104

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

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

Step 8.3.105

Multiply $4$ by $2$ .

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

Step 8.3.106

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

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

Step 8.3.107

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.108

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.109

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

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

Step 8.3.110

Add $1$ and $1$ .

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

Step 8.3.111

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

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

Step 8.3.112

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

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

Step 8.3.113

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

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

Step 8.3.114

Multiply $2$ by $2$ .

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

Step 8.3.115

Combine $\frac{\cos (u_{2})}{4}$ and $\cos (u_{2})$ .

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

Step 8.3.116

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.117

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.118

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

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

Step 8.3.119

Add $1$ and $1$ .

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

Step 8.3.120

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

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

Step 8.3.121

Multiply $4$ by $2$ .

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

Step 8.3.122

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

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

Step 8.3.123

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

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

Step 8.3.124

Multiply $2$ by $2$ .

Step 8.3.125

Combine $\frac{\cos (u_{2})}{4}$ and $\cos (u_{2})$ .

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

Step 8.3.126

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.127

Raise $\cos (u_{2})$ to the power of $1$ .

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

Step 8.3.128

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

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

Step 8.3.129

Add $1$ and $1$ .

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

Step 8.3.130

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

Step 8.3.131

Multiply $4$ by $2$ .

Step 8.3.132

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

Step 8.3.133

Raise $\cos (u_{2})$ to the power of $1$ .

Step 8.3.134

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

Step 8.3.135

Add $2$ and $1$ .

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

Step 8.3.136

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

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

Step 8.3.137

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

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

Step 8.3.138

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

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

Step 8.3.139

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

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

Step 8.3.140

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

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

Step 8.3.141

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

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

Step 8.3.142

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

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

Step 8.3.143

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

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

Step 8.3.144

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

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

Step 8.3.145

Combine the numerators over the common denominator.

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

Step 8.3.146

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

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

Step 8.3.147

Combine the numerators over the common denominator.

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

Step 8.3.148

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

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Since $\frac{1}{8}$ is constant with respect to $u_{2}$ , move $\frac{1}{8}$ out of the integral.

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

Step 11

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

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

Step 12

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

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

Step 13

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

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Step 13.1

Let $u_{3} = \sin (u_{2})$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 13.1.1

Differentiate $\sin (u_{2})$ .

$\frac{d}{d u_{2}} [\sin (u_{2})]$

Step 13.1.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\cos (u_{2})$

Step 13.2

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

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

Step 14

Split the single integral into multiple integrals.

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

Step 15

Apply the constant rule.

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

Step 16

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

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

Step 17

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

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

Step 18

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

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

Step 19

Since $\frac{3}{8}$ is constant with respect to $u_{2}$ , move $\frac{3}{8}$ out of the integral.

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

Step 20

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

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

Step 21

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

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

Step 22

Simplify.

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Step 22.1

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

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

Step 22.2

Multiply $2$ by $8$ .

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

Step 23

Split the single integral into multiple integrals.

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

Step 24

Apply the constant rule.

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

Step 25

Let $u_{4} = 2 u_{2}$ . Then $d u_{4} = 2 d u_{2}$ , so $\frac{1}{2} d u_{4} = d u_{2}$ . Rewrite using $u_{4}$ and $d$ $u_{4}$ .

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Step 25.1

Let $u_{4} = 2 u_{2}$ . Find $\frac{d u_{4}}{d u_{2}}$ .

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Step 25.1.1

Differentiate $2 u_{2}$ .

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

Step 25.1.2

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

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

Step 25.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 25.1.4

Multiply $2$ by $1$ .

$2$

Step 25.2

Rewrite the problem using $u_{4}$ and $d u_{4}$ .

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \int \cos (u_{4}) \frac{1}{2} d u_{4}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

Step 26

Combine $\cos (u_{4})$ and $\frac{1}{2}$ .

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \int \frac{\cos (u_{4})}{2} d u_{4}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

Step 27

Since $\frac{1}{2}$ is constant with respect to $u_{4}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} \int \cos (u_{4}) d u_{4}) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

Step 28

The integral of $\cos (u_{4})$ with respect to $u_{4}$ is $\sin (u_{4})$ .

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \int \frac{1}{8} d u_{2} + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

Step 29

Apply the constant rule.

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \frac{1}{8} u_{2} + C + \int \frac{3 \cos (u_{2})}{8} d u_{2}$

Step 30

Since $\frac{3}{8}$ is constant with respect to $u_{2}$ , move $\frac{3}{8}$ out of the integral.

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \frac{1}{8} u_{2} + C + \frac{3}{8} \int \cos (u_{2}) d u_{2}$

Step 31

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{8} (u_{3} + C - (\frac{{u_{3}}^{3}}{3} + C)) + \frac{3}{16} (u_{2} + C + \frac{1}{2} (\sin (u_{4}) + C)) + \frac{1}{8} u_{2} + C + \frac{3}{8} (\sin (u_{2}) + C)$

Step 32

Simplify.

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

Step 33

Reorder terms.

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

Step 34

Substitute back in for each integration substitution variable.

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Step 34.1

Replace all occurrences of $u_{3}$ with $\sin (u_{2})$ .

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

Step 34.2

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

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

Step 34.3

Replace all occurrences of $u_{4}$ with $2 u_{2}$ .

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

Step 34.4

Replace all occurrences of $u_{1}$ with $\frac{y}{2}$ .

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

Step 34.5

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

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

Step 34.6

Replace all occurrences of $u_{1}$ with $\frac{y}{2}$ .

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

Step 35

Simplify.

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Step 35.1

Combine $2$ and $\frac{y}{2}$ .

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

Step 35.2

Combine $2$ and $\frac{y}{2}$ .

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

Step 35.3

Combine $\sin^{3} (\frac{2 y}{2})$ and $\frac{1}{24}$ .

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

Step 35.4

Combine $2$ and $\frac{y}{2}$ .

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

Step 35.5

Multiply $\frac{3}{16}$ by $\frac{2 y}{2}$ .

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

Step 35.6

Multiply $2$ by $3$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{16 \cdot 2} + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.7

Multiply $16$ by $2$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (2 (2 \frac{y}{2})) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.8

Combine $2$ and $\frac{y}{2}$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (2 \frac{2 y}{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.9

Combine $2$ and $\frac{2 y}{2}$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{2 (2 y)}{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.10

Multiply $2$ by $2$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{1}{8} (2 \frac{y}{2}) + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.11

Combine $2$ and $\frac{y}{2}$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{1}{8} \cdot \frac{2 y}{2} + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.12

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

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{8 \cdot 2} + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.13

Multiply $8$ by $2$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (2 \frac{y}{2}) + C$

Step 35.14

Combine $2$ and $\frac{y}{2}$ .

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

Step 36

Simplify.

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Step 36.1

Cancel the common factor of $2$ .

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Step 36.1.1

Cancel the common factor.

$\frac{1}{8} \sin (\frac{2 y}{2}) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

Step 36.1.2

Divide $y$ by $1$ .

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

Step 36.2

Cancel the common factor of $2$ .

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Step 36.2.1

Cancel the common factor.

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (\frac{2 y}{2})}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

Step 36.2.2

Divide $y$ by $1$ .

$\frac{1}{8} \sin (y) - \frac{\sin^{3} (y)}{24} + \frac{6 y}{32} + \frac{3}{32} \sin (\frac{4 y}{2}) + \frac{2 y}{16} + \frac{3}{8} \sin (\frac{2 y}{2}) + C$

Step 36.3

Cancel the common factor of $6$ and $32$ .

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Step 36.3.1

Factor $2$ out of $6 y$ .

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

Step 36.3.2

Cancel the common factors.

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Step 36.3.2.1

Factor $2$ out of $32$ .

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

Step 36.3.2.2

Cancel the common factor.

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

Step 36.3.2.3

Rewrite the expression.

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

Step 36.4

Cancel the common factor of $4$ and $2$ .

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Step 36.4.1

Factor $2$ out of $4 y$ .

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

Step 36.4.2

Cancel the common factors.

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Step 36.4.2.1

Factor $2$ out of $2$ .

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

Step 36.4.2.2

Cancel the common factor.

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

Step 36.4.2.3

Rewrite the expression.

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

Step 36.4.2.4

Divide $2 y$ by $1$ .

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

Step 36.5

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

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Step 36.5.1

Factor $2$ out of $2 y$ .

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

Step 36.5.2

Cancel the common factors.

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Step 36.5.2.1

Factor $2$ out of $16$ .

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

Step 36.5.2.2

Cancel the common factor.

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

Step 36.5.2.3

Rewrite the expression.

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

Step 36.6

Cancel the common factor of $2$ .

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Step 36.6.1

Cancel the common factor.

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

Step 36.6.2

Divide $y$ by $1$ .

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

Step 36.7

Reorder terms.

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