Calculus Examples

$\cos^{4} (2 x)$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify with factoring out.

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Step 4.1

Factor $2$ out of $4$ .

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

Step 4.2

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

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

Step 5

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

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

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 $d u_{2}$ .

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

Step 7

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

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

Step 8

Simplify terms.

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Step 8.1

Simplify.

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Step 8.1.1

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

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

Step 8.1.2

Multiply $2$ by $2$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

Rewrite the exponentiation as a product.

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

Step 8.3.2

Apply the distributive property.

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

Step 8.3.3

Apply the distributive property.

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

Step 8.3.4

Apply the distributive property.

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

Step 8.3.5

Apply the distributive property.

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

Step 8.3.6

Apply the distributive property.

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

Step 8.3.7

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

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

Step 8.3.8

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

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

Step 8.3.9

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

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

Step 8.3.10

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

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

Step 8.3.11

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

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

Step 8.3.12

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

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

Step 8.3.13

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

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

Step 8.3.14

Multiply $1$ by $1$ .

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

Step 8.3.15

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

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

Step 8.3.16

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

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

Step 8.3.17

Multiply $2$ by $2$ .

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

Step 8.3.18

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

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

Step 8.3.19

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

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

Step 8.3.20

Multiply $2$ by $2$ .

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

Step 8.3.21

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

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

Step 8.3.22

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

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

Step 8.3.23

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

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

Step 8.3.24

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

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

Step 8.3.25

Multiply $2$ by $2$ .

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

Step 8.3.26

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

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

Step 8.3.27

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

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

Step 8.3.28

Multiply $2$ by $2$ .

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

Step 8.3.29

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

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

Step 8.3.30

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

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

Step 8.3.31

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

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

Step 8.3.32

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

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

Step 8.3.33

Add $1$ and $1$ .

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

Step 8.3.34

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

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

Step 8.3.35

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

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

Step 8.3.36

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

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

Step 8.3.37

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

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

Step 8.4

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

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Step 8.4.1

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

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

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $2$ out of $4$ .

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

Step 8.4.2.2

Cancel the common factor.

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

Step 8.4.2.3

Rewrite the expression.

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Multiply $2$ by $4$ .

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

Step 14

Split the single integral into multiple integrals.

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

Step 15

Apply the constant rule.

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

Step 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 16.1

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

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Step 16.1.1

Differentiate $2 u_{2}$ .

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

Step 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 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 16.1.4

Multiply $2$ by $1$ .

$2$

Step 16.2

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

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

Step 17

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

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

Step 18

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

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

Step 19

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

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

Step 20

Apply the constant rule.

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

Step 21

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

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

Step 22

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

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

Step 23

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

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

Step 24

Simplify.

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Step 24.1

Simplify.

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

Step 24.2

Simplify.

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Step 24.2.1

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

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

Step 24.2.2

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

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Step 24.2.2.1

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

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

Step 24.2.2.2

Multiply $4$ by $2$ .

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

Step 24.2.3

Combine the numerators over the common denominator.

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

Step 24.2.4

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

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

Step 24.2.5

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

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

Step 25

Substitute back in for each integration substitution variable.

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Step 25.1

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

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

Step 25.2

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

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

Step 25.3

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

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

Step 25.4

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

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

Step 25.5

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

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

Step 26

Simplify.

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Step 26.1

Simplify each term.

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Step 26.1.1

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

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Step 26.1.1.1

Factor $2$ out of $3 (2 (2 x))$ .

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

Step 26.1.1.2

Cancel the common factors.

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Step 26.1.1.2.1

Factor $2$ out of $8$ .

$\frac{1}{4} (\frac{2 (3 (2 x))}{2 \cdot 4} + \frac{\sin (2 (2 (2 x)))}{16} + \frac{\sin (2 (2 x))}{2}) + C$

Step 26.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{2 (3 (2 x))}{2 \cdot 4} + \frac{\sin (2 (2 (2 x)))}{16} + \frac{\sin (2 (2 x))}{2}) + C$

Step 26.1.1.2.3

Rewrite the expression.

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

Step 26.1.2

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

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Step 26.1.2.1

Factor $2$ out of $3 (2 x)$ .

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

Step 26.1.2.2

Cancel the common factors.

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Step 26.1.2.2.1

Factor $2$ out of $4$ .

$\frac{1}{4} (\frac{2 (3 (x))}{2 \cdot 2} + \frac{\sin (2 (2 (2 x)))}{16} + \frac{\sin (2 (2 x))}{2}) + C$

Step 26.1.2.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{2 (3 (x))}{2 \cdot 2} + \frac{\sin (2 (2 (2 x)))}{16} + \frac{\sin (2 (2 x))}{2}) + C$

Step 26.1.2.2.3

Rewrite the expression.

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

Step 26.1.3

Multiply $2 \cdot 2 \cdot 2$ .

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Step 26.1.3.1

Multiply $2$ by $2$ .

$\frac{1}{4} (\frac{3 x}{2} + \frac{\sin (4 \cdot 2 x)}{16} + \frac{\sin (2 (2 x))}{2}) + C$

Step 26.1.3.2

Multiply $4$ by $2$ .

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

Step 26.1.4

Multiply $2$ by $2$ .

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

Step 26.2

Apply the distributive property.

$\frac{1}{4} \cdot \frac{3 x}{2} + \frac{1}{4} \cdot \frac{\sin (8 x)}{16} + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 26.3

Simplify.

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Step 26.3.1

Multiply $\frac{1}{4} \cdot \frac{3 x}{2}$ .

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Step 26.3.1.1

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

$\frac{3 x}{4 \cdot 2} + \frac{1}{4} \cdot \frac{\sin (8 x)}{16} + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 26.3.1.2

Multiply $4$ by $2$ .

$\frac{3 x}{8} + \frac{1}{4} \cdot \frac{\sin (8 x)}{16} + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 26.3.2

Multiply $\frac{1}{4} \cdot \frac{\sin (8 x)}{16}$ .

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Step 26.3.2.1

Multiply $\frac{1}{4}$ by $\frac{\sin (8 x)}{16}$ .

$\frac{3 x}{8} + \frac{\sin (8 x)}{4 \cdot 16} + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 26.3.2.2

Multiply $4$ by $16$ .

$\frac{3 x}{8} + \frac{\sin (8 x)}{64} + \frac{1}{4} \cdot \frac{\sin (4 x)}{2} + C$

Step 26.3.3

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

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Step 26.3.3.1

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

$\frac{3 x}{8} + \frac{\sin (8 x)}{64} + \frac{\sin (4 x)}{4 \cdot 2} + C$

Step 26.3.3.2

Multiply $4$ by $2$ .

$\frac{3 x}{8} + \frac{\sin (8 x)}{64} + \frac{\sin (4 x)}{8} + C$

Step 27

Reorder terms.

$\frac{3}{8} x + \frac{1}{64} \sin (8 x) + \frac{1}{8} \sin (4 x) + C$