Calculus Examples

$\frac{d r}{d θ} = \sin^{4} (π θ)$

Step 1

Rewrite the equation.

$d r = \sin^{4} (π θ) d θ$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d r = \int \sin^{4} (π θ) d θ$

Step 2.2

Apply the constant rule.

$r + C_{1} = \int \sin^{4} (π θ) d θ$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $π θ$ .

$\frac{d}{d θ} [π θ]$

Step 2.3.1.1.2

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

$π \frac{d}{d θ} [θ]$

Step 2.3.1.1.3

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

$π \cdot 1$

Step 2.3.1.1.4

Multiply $π$ by $1$ .

$π$

Step 2.3.1.2

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

$r + C_{1} = \int \sin^{4} (u_{1}) \frac{1}{π} d u_{1}$

Step 2.3.2

Combine $\sin^{4} (u_{1})$ and $\frac{1}{π}$ .

$r + C_{1} = \int \frac{\sin^{4} (u_{1})}{π} d u_{1}$

Step 2.3.3

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

$r + C_{1} = \frac{1}{π} \int \sin^{4} (u_{1}) d u_{1}$

Step 2.3.4

Simplify with factoring out.

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Step 2.3.4.1

Factor $2$ out of $4$ .

$r + C_{1} = \frac{1}{π} \int {\sin (u_{1})}^{2 (2)} d u_{1}$

Step 2.3.4.2

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

$r + C_{1} = \frac{1}{π} \int {(\sin^{2} (u_{1}))}^{2} d u_{1}$

Step 2.3.5

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

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

Step 2.3.6

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 2.3.6.1

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

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Step 2.3.6.1.1

Differentiate $2 u_{1}$ .

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

Step 2.3.6.1.2

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 2.3.6.1.3

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 2.3.6.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.6.2

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

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

Step 2.3.7

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

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

Step 2.3.8

Simplify terms.

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Step 2.3.8.1

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

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

Step 2.3.8.2

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

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

Step 2.3.8.3

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

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Step 2.3.8.3.1

Rewrite the exponentiation as a product.

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{2} \cdot (1 - \cos (u_{2})) (\frac{1}{2} \cdot (1 - \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.2

Apply the distributive property.

$r + C_{1} = \frac{1}{2 π} \int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{2}))) (\frac{1}{2} \cdot (1 - \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.3

Apply the distributive property.

$r + C_{1} = \frac{1}{2 π} \int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{2}))) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.4

Apply the distributive property.

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{2}))) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.5

Apply the distributive property.

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{2}))) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.6

Apply the distributive property.

Step 2.3.8.3.7

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{2}))) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.8

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{2}))) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.9

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{2}))) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.10

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot (- \cos (u_{2}))) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.11

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2} \cos (u_{2})) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.12

Move parentheses.

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2}) \cos (u_{2}) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.13

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.14

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot \frac{1}{2} \cos (u_{2}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.15

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot \frac{1}{2} \cos (u_{2}) (1 \cdot \frac{1}{2}) + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.16

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot \frac{1}{2} \cdot 1 \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.17

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} + \frac{1}{2} \cdot (- \cos (u_{2})) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.18

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cos (u_{2}) (\frac{1}{2} \cdot (- \cos (u_{2}))) d u_{2}$

Step 2.3.8.3.19

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cos (u_{2}) (- 1 \cdot \frac{1}{2} \cos (u_{2})) d u_{2}$

Step 2.3.8.3.20

Move parentheses.

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cos (u_{2}) (- 1 \cdot \frac{1}{2}) \cos (u_{2}) d u_{2}$

Step 2.3.8.3.21

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cdot - 1 \cos (u_{2}) \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.22

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

$r + C_{1} = \frac{1}{2 π} \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{2}) \cdot \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.23

Multiply $1$ by $1$ .

$r + C_{1} = \frac{1}{2 π} \int 1 (\frac{1}{2}) \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.24

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

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.25

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

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{2 \cdot 2} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{2}) - 1 \cdot 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.26

Multiply $2$ by $2$ .

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

Step 2.3.8.3.27

Multiply $- 1$ by $1$ .

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

Step 2.3.8.3.28

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

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

Step 2.3.8.3.29

Multiply $2$ by $2$ .

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

Step 2.3.8.3.30

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

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

Step 2.3.8.3.31

Multiply $- 1$ by $1$ .

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

Step 2.3.8.3.32

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

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

Step 2.3.8.3.33

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

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

Step 2.3.8.3.34

Multiply $2$ by $2$ .

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

Step 2.3.8.3.35

Multiply $- 1$ by $- 1$ .

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{4} - \frac{\cos (u_{2})}{4} - \frac{\cos (u_{2})}{4} + 1 (\frac{1}{2}) \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.36

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

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{4} - \frac{\cos (u_{2})}{4} - \frac{\cos (u_{2})}{4} + \frac{1}{2} \cos (u_{2}) \frac{1}{2} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.37

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

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

Step 2.3.8.3.38

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

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

Step 2.3.8.3.39

Multiply $2$ by $2$ .

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{4} - \frac{\cos (u_{2})}{4} - \frac{\cos (u_{2})}{4} + \frac{\cos (u_{2})}{4} \cos (u_{2}) d u_{2}$

Step 2.3.8.3.40

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

$r + C_{1} = \frac{1}{2 π} \int \frac{1}{4} - \frac{\cos (u_{2})}{4} - \frac{\cos (u_{2})}{4} + \frac{\cos (u_{2}) \cos (u_{2})}{4} d u_{2}$

Step 2.3.8.3.41

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

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

Step 2.3.8.3.42

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

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

Step 2.3.8.3.43

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

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

Step 2.3.8.3.44

Add $1$ and $1$ .

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

Step 2.3.8.3.45

Subtract $\frac{\cos (u_{2})}{4}$ from $- \frac{\cos (u_{2})}{4}$ .

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

Step 2.3.8.3.46

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

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

Step 2.3.8.3.47

Reorder $\frac{- 2 \cos (u_{2})}{4}$ and $\frac{\cos^{2} (u_{2})}{4}$ .

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

Step 2.3.8.3.48

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

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

Step 2.3.8.4

Simplify.

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Step 2.3.8.4.1

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

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Step 2.3.8.4.1.1

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

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

Step 2.3.8.4.1.2

Cancel the common factors.

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Step 2.3.8.4.1.2.1

Factor $2$ out of $4$ .

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

Step 2.3.8.4.1.2.2

Cancel the common factor.

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

Step 2.3.8.4.1.2.3

Rewrite the expression.

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

Step 2.3.8.4.2

Move the negative in front of the fraction.

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

Step 2.3.9

Split the single integral into multiple integrals.

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

Step 2.3.10

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

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

Step 2.3.11

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{4} \int \frac{1 + \cos (2 u_{2})}{2} d u_{2} + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.12

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{4} (\frac{1}{2} \int 1 + \cos (2 u_{2}) d u_{2}) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.13

Simplify.

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Step 2.3.13.1

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

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

Step 2.3.13.2

Multiply $2$ by $4$ .

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} \int 1 + \cos (2 u_{2}) d u_{2} + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.14

Split the single integral into multiple integrals.

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (\int d u_{2} + \int \cos (2 u_{2}) d u_{2}) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.15

Apply the constant rule.

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \int \cos (2 u_{2}) d u_{2}) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.16

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

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Step 2.3.16.1

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

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Step 2.3.16.1.1

Differentiate $2 u_{2}$ .

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

Step 2.3.16.1.2

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 2.3.16.1.3

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 2.3.16.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.16.2

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \int \cos (u_{3}) \frac{1}{2} d u_{3}) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.17

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \int \frac{\cos (u_{3})}{2} d u_{3}) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.18

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} \int \cos (u_{3}) d u_{3}) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.19

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} (\sin (u_{3}) + C_{3})) + \int \frac{1}{4} d u_{2} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.20

Apply the constant rule.

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} (\sin (u_{3}) + C_{3})) + \frac{1}{4} u_{2} + C_{4} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.21

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} (\sin (u_{3}) + C_{3})) + \frac{u_{2}}{4} + C_{4} + \int - \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.22

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} (\sin (u_{3}) + C_{3})) + \frac{u_{2}}{4} + C_{4} - \int \frac{\cos (u_{2})}{2} d u_{2})$

Step 2.3.23

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} (\sin (u_{3}) + C_{3})) + \frac{u_{2}}{4} + C_{4} - (\frac{1}{2} \int \cos (u_{2}) d u_{2}))$

Step 2.3.24

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

$r + C_{1} = \frac{1}{2 π} (\frac{1}{8} (u_{2} + C_{2} + \frac{1}{2} (\sin (u_{3}) + C_{3})) + \frac{u_{2}}{4} + C_{4} - \frac{1}{2} (\sin (u_{2}) + C_{5}))$

Step 2.3.25

Simplify.

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Step 2.3.25.1

Simplify.

$r + C_{1} = \frac{1}{2 π} (\frac{u_{2}}{8} + \frac{\sin (u_{3})}{16} + \frac{u_{2}}{4} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.25.2

Simplify.

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Step 2.3.25.2.1

To write $\frac{u_{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r + C_{1} = \frac{1}{2 π} (\frac{u_{2}}{8} + \frac{u_{2}}{4} \cdot \frac{2}{2} + \frac{\sin (u_{3})}{16} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.25.2.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.25.2.2.1

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

$r + C_{1} = \frac{1}{2 π} (\frac{u_{2}}{8} + \frac{u_{2} \cdot 2}{4 \cdot 2} + \frac{\sin (u_{3})}{16} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.25.2.2.2

Multiply $4$ by $2$ .

$r + C_{1} = \frac{1}{2 π} (\frac{u_{2}}{8} + \frac{u_{2} \cdot 2}{8} + \frac{\sin (u_{3})}{16} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.25.2.3

Combine the numerators over the common denominator.

$r + C_{1} = \frac{1}{2 π} (\frac{u_{2} + u_{2} \cdot 2}{8} + \frac{\sin (u_{3})}{16} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.25.2.4

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

$r + C_{1} = \frac{1}{2 π} (\frac{u_{2} + 2 \cdot u_{2}}{8} + \frac{\sin (u_{3})}{16} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.25.2.5

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

$r + C_{1} = \frac{1}{2 π} (\frac{3 u_{2}}{8} + \frac{\sin (u_{3})}{16} - \frac{\sin (u_{2})}{2}) + C_{6}$

Step 2.3.26

Substitute back in for each integration substitution variable.

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Step 2.3.26.1

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

$r + C_{1} = \frac{1}{2 π} (\frac{3 (2 u_{1})}{8} + \frac{\sin (u_{3})}{16} - \frac{\sin (2 u_{1})}{2}) + C_{6}$

Step 2.3.26.2

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

$r + C_{1} = \frac{1}{2 π} (\frac{3 (2 u_{1})}{8} + \frac{\sin (2 u_{2})}{16} - \frac{\sin (2 u_{1})}{2}) + C_{6}$

Step 2.3.26.3

Replace all occurrences of $u_{1}$ with $π θ$ .

$r + C_{1} = \frac{1}{2 π} (\frac{3 (2 (π θ))}{8} + \frac{\sin (2 u_{2})}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.26.4

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

$r + C_{1} = \frac{1}{2 π} (\frac{3 (2 (π θ))}{8} + \frac{\sin (2 (2 u_{1}))}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.26.5

Replace all occurrences of $u_{1}$ with $π θ$ .

$r + C_{1} = \frac{1}{2 π} (\frac{3 (2 (π θ))}{8} + \frac{\sin (2 (2 (π θ)))}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.27

Simplify.

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Step 2.3.27.1

Simplify each term.

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Step 2.3.27.1.1

Cancel the common factor of $2$ and $8$ .

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Step 2.3.27.1.1.1

Factor $2$ out of $3 (2 (π θ))$ .

$r + C_{1} = \frac{1}{2 π} (\frac{2 (3 (π θ))}{8} + \frac{\sin (2 (2 (π θ)))}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.27.1.1.2

Cancel the common factors.

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Step 2.3.27.1.1.2.1

Factor $2$ out of $8$ .

$r + C_{1} = \frac{1}{2 π} (\frac{2 (3 (π θ))}{2 \cdot 4} + \frac{\sin (2 (2 (π θ)))}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.27.1.1.2.2

Cancel the common factor.

$r + C_{1} = \frac{1}{2 π} (\frac{2 (3 (π θ))}{2 \cdot 4} + \frac{\sin (2 (2 (π θ)))}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.27.1.1.2.3

Rewrite the expression.

$r + C_{1} = \frac{1}{2 π} (\frac{3 (π θ)}{4} + \frac{\sin (2 (2 (π θ)))}{16} - \frac{\sin (2 (π θ))}{2}) + C_{6}$

Step 2.3.27.1.2

Multiply $2$ by $2$ .

$r + C_{1} = \frac{1}{2 π} (\frac{3 π θ}{4} + \frac{\sin (4 π θ)}{16} - \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.2

Apply the distributive property.

$r + C_{1} = \frac{1}{2 π} \cdot \frac{3 π θ}{4} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3

Simplify.

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Step 2.3.27.3.1

Cancel the common factor of $π$ .

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Step 2.3.27.3.1.1

Factor $π$ out of $2 π$ .

$r + C_{1} = \frac{1}{π \cdot 2} \cdot \frac{3 π θ}{4} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.1.2

Factor $π$ out of $3 π θ$ .

$r + C_{1} = \frac{1}{π \cdot 2} \cdot \frac{π (3 θ)}{4} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.1.3

Cancel the common factor.

$r + C_{1} = \frac{1}{π \cdot 2} \cdot \frac{π (3 θ)}{4} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.1.4

Rewrite the expression.

$r + C_{1} = \frac{1}{2} \cdot \frac{3 θ}{4} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.2

Multiply $\frac{1}{2}$ by $\frac{3 θ}{4}$ .

$r + C_{1} = \frac{3 θ}{2 \cdot 4} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.3

Multiply $2$ by $4$ .

$r + C_{1} = \frac{3 θ}{8} + \frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.4

Multiply $\frac{1}{2 π} \cdot \frac{\sin (4 π θ)}{16}$ .

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Step 2.3.27.3.4.1

Multiply $\frac{1}{2 π}$ by $\frac{\sin (4 π θ)}{16}$ .

$r + C_{1} = \frac{3 θ}{8} + \frac{\sin (4 π θ)}{2 π \cdot 16} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.4.2

Multiply $16$ by $2$ .

$r + C_{1} = \frac{3 θ}{8} + \frac{\sin (4 π θ)}{32 π} + \frac{1}{2 π} (- \frac{\sin (2 π θ)}{2}) + C_{6}$

Step 2.3.27.3.5

Multiply $\frac{1}{2 π} (- \frac{\sin (2 π θ)}{2})$ .

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Step 2.3.27.3.5.1

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

$r + C_{1} = \frac{3 θ}{8} + \frac{\sin (4 π θ)}{32 π} - \frac{\sin (2 π θ)}{2 π \cdot 2} + C_{6}$

Step 2.3.27.3.5.2

Multiply $2$ by $2$ .

$r + C_{1} = \frac{3 θ}{8} + \frac{\sin (4 π θ)}{32 π} - \frac{\sin (2 π θ)}{4 π} + C_{6}$

Step 2.3.28

Reorder terms.

$r + C_{1} = \frac{3}{8} θ + \frac{1}{32 π} \sin (4 π θ) - \frac{1}{4 π} \sin (2 π θ) + C_{6}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$r = \frac{3}{8} θ + \frac{1}{32 π} \sin (4 π θ) - \frac{1}{4 π} \sin (2 π θ) + K$