Calculus Examples

$\cos^{4} (x) - \sin^{4} (x)$

Step 1

Write $\cos^{4} (x) - \sin^{4} (x)$ as a function.

$f (x) = \cos^{4} (x) - \sin^{4} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \cos^{4} (x) - \sin^{4} (x) d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Simplify with factoring out.

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Step 5.1

Factor $2$ out of $4$ .

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

Step 5.2

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

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

Step 6

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

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

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $2 x$ .

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

Step 7.1.2

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 7.1.3

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 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

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

Step 8

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

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

Step 9

Simplify terms.

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

Rewrite the exponentiation as a product.

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

Step 9.2.2

Apply the distributive property.

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

Step 9.2.3

Apply the distributive property.

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

Step 9.2.4

Apply the distributive property.

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

Step 9.2.5

Apply the distributive property.

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

Step 9.2.6

Apply the distributive property.

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

Step 9.2.7

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

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

Step 9.2.8

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

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

Step 9.2.9

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

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

Step 9.2.10

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

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

Step 9.2.11

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

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

Step 9.2.12

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

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

Step 9.2.13

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

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

Step 9.2.14

Multiply $1$ by $1$ .

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

Step 9.2.15

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

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

Step 9.2.16

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

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

Step 9.2.17

Multiply $2$ by $2$ .

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

Step 9.2.18

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

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

Step 9.2.19

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

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

Step 9.2.20

Multiply $2$ by $2$ .

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

Step 9.2.21

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

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

Step 9.2.22

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

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

Step 9.2.23

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

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

Step 9.2.24

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

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

Step 9.2.25

Multiply $2$ by $2$ .

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

Step 9.2.26

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

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

Step 9.2.27

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

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

Step 9.2.28

Multiply $2$ by $2$ .

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

Step 9.2.29

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

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

Step 9.2.30

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

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

Step 9.2.31

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

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

Step 9.2.32

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

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

Step 9.2.33

Add $1$ and $1$ .

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

Step 9.2.34

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

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

Step 9.2.35

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

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

Step 9.2.36

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

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

Step 9.2.37

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $4$ .

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

Step 9.3.2.2

Cancel the common factor.

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

Step 9.3.2.3

Rewrite the expression.

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify.

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Step 14.1

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

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

Step 14.2

Multiply $2$ by $4$ .

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

Step 15

Split the single integral into multiple integrals.

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

Step 16

Apply the constant rule.

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

Step 17

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 17.1

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

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Step 17.1.1

Differentiate $2 u_{1}$ .

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

Step 17.1.2

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 17.1.3

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 17.1.4

Multiply $2$ by $1$ .

$2$

Step 17.2

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

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

Step 18

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

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

Step 19

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

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

Step 20

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

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

Step 21

Apply the constant rule.

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

Step 22

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

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

Step 23

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

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

Step 24

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

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

Step 25

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

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

Step 26

Simplify with factoring out.

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Step 26.1

Factor $2$ out of $4$ .

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

Step 26.2

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

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

Step 27

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

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

Step 28

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

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Step 28.1

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

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Step 28.1.1

Differentiate $2 x$ .

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

Step 28.1.2

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 28.1.3

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 28.1.4

Multiply $2$ by $1$ .

$2$

Step 28.2

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

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

Step 29

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

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

Step 30

Simplify by multiplying through.

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Step 30.1

Rewrite $\frac{1 - \cos (u_{3})}{2}$ as a product.

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

Step 30.2

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

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Step 30.2.1

Rewrite the exponentiation as a product.

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

Step 30.2.2

Apply the distributive property.

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

Step 30.2.3

Apply the distributive property.

Step 30.2.4

Apply the distributive property.

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

Step 30.2.5

Apply the distributive property.

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

Step 30.2.6

Apply the distributive property.

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

Step 30.2.7

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

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

Step 30.2.8

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

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

Step 30.2.9

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

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

Step 30.2.10

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

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

Step 30.2.11

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

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

Step 30.2.12

Move parentheses.

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

Step 30.2.13

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

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

Step 30.2.14

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

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

Step 30.2.15

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

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

Step 30.2.16

Move $\cos (u_{3})$ .

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

Step 30.2.17

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

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

Step 30.2.18

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

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

Step 30.2.19

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

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

Step 30.2.20

Move parentheses.

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

Step 30.2.21

Move $\cos (u_{3})$ .

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

Step 30.2.22

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

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

Step 30.2.23

Multiply $1$ by $1$ .

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

Step 30.2.24

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

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

Step 30.2.25

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

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

Step 30.2.26

Multiply $2$ by $2$ .

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

Step 30.2.27

Multiply $- 1$ by $1$ .

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

Step 30.2.28

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

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

Step 30.2.29

Multiply $2$ by $2$ .

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

Step 30.2.30

Combine $\frac{1}{4}$ and $\cos (u_{3})$ .

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

Step 30.2.31

Multiply $- 1$ by $1$ .

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

Step 30.2.32

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

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

Step 30.2.33

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

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

Step 30.2.34

Multiply $2$ by $2$ .

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

Step 30.2.35

Multiply $- 1$ by $- 1$ .

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

Step 30.2.36

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

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

Step 30.2.37

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

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

Step 30.2.38

Multiply $\frac{\cos (u_{3})}{2}$ by $\frac{1}{2}$ .

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

Step 30.2.39

Multiply $2$ by $2$ .

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

Step 30.2.40

Combine $\frac{\cos (u_{3})}{4}$ and $\cos (u_{3})$ .

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

Step 30.2.41

Raise $\cos (u_{3})$ to the power of $1$ .

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

Step 30.2.42

Raise $\cos (u_{3})$ to the power of $1$ .

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

Step 30.2.43

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

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

Step 30.2.44

Add $1$ and $1$ .

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

Step 30.2.45

Subtract $\frac{\cos (u_{3})}{4}$ from $- \frac{\cos (u_{3})}{4}$ .

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

Step 30.2.46

Combine $- 2$ and $\frac{\cos (u_{3})}{4}$ .

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

Step 30.2.47

Reorder $\frac{- 2 \cos (u_{3})}{4}$ and $\frac{\cos^{2} (u_{3})}{4}$ .

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

Step 30.2.48

Reorder $\frac{1}{4}$ and $\frac{\cos^{2} (u_{3})}{4}$ .

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

Step 30.3

Simplify.

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Step 30.3.1

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

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Step 30.3.1.1

Factor $2$ out of $- 2 \cos (u_{3})$ .

Step 30.3.1.2

Cancel the common factors.

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Step 30.3.1.2.1

Factor $2$ out of $4$ .

Step 30.3.1.2.2

Cancel the common factor.

Step 30.3.1.2.3

Rewrite the expression.

Step 30.3.2

Move the negative in front of the fraction.

Step 31

Split the single integral into multiple integrals.

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

Step 32

Since $\frac{1}{4}$ is constant with respect to $u_{3}$ , move $\frac{1}{4}$ out of the integral.

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

Step 33

Use the half-angle formula to rewrite $\cos^{2} (u_{3})$ as $\frac{1 + \cos (2 u_{3})}{2}$ .

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

Step 34

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

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

Step 35

Simplify.

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Step 35.1

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

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

Step 35.2

Multiply $2$ by $4$ .

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

Step 36

Split the single integral into multiple integrals.

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

Step 37

Apply the constant rule.

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

Step 38

Let $u_{4} = 2 u_{3}$ . Then $d u_{4} = 2 d u_{3}$ , so $\frac{1}{2} d u_{4} = d u_{3}$ . Rewrite using $u_{4}$ and $d$ $u_{4}$ .

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Step 38.1

Let $u_{4} = 2 u_{3}$ . Find $\frac{d u_{4}}{d u_{3}}$ .

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Step 38.1.1

Differentiate $2 u_{3}$ .

$\frac{d}{d u_{3}} [2 u_{3}]$

Step 38.1.2

Since $2$ is constant with respect to $u_{3}$ , the derivative of $2 u_{3}$ with respect to $u_{3}$ is $2 \frac{d}{d u_{3}} [u_{3}]$ .

$2 \frac{d}{d u_{3}} [u_{3}]$

Step 38.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 38.1.4

Multiply $2$ by $1$ .

$2$

Step 38.2

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

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

Step 39

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

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

Step 40

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

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

Step 41

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

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

Step 42

Apply the constant rule.

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

Step 43

Combine $\frac{1}{4}$ and $u_{3}$ .

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

Step 44

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

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

Step 45

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

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

Step 46

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

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

Step 47

Simplify.

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Step 47.1

Simplify.

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

Step 47.2

Simplify.

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Step 47.2.1

To write $\frac{u_{1}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 47.2.2

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

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Step 47.2.2.1

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

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

Step 47.2.2.2

Multiply $4$ by $2$ .

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

Step 47.2.3

Combine the numerators over the common denominator.

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

Step 47.2.4

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

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

Step 47.2.5

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

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

Step 47.2.6

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

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

Step 47.2.7

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

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

Step 47.2.8

To write $\frac{1}{8} (u_{3} + \frac{\sin (u_{4})}{2})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 47.2.9

Combine $\frac{1}{8} (u_{3} + \frac{\sin (u_{4})}{2})$ and $\frac{2}{2}$ .

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

Step 47.2.10

Combine the numerators over the common denominator.

Step 47.2.11

Combine $2$ and $\frac{1}{8}$ .

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

Step 47.2.12

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

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Step 47.2.12.1

Factor $2$ out of $2$ .

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

Step 47.2.12.2

Cancel the common factors.

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Step 47.2.12.2.1

Factor $2$ out of $8$ .

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

Step 47.2.12.2.2

Cancel the common factor.

Step 47.2.12.2.3

Rewrite the expression.

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

Step 48

Substitute back in for each integration substitution variable.

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Step 48.1

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

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

Step 48.2

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

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

Step 48.3

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

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

Step 48.4

Replace all occurrences of $u_{4}$ with $2 u_{3}$ .

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

Step 48.5

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

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

Step 48.6

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

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

Step 49

Simplify.

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Step 49.1

Simplify each term.

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Step 49.1.1

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

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Step 49.1.1.1

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

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

Step 49.1.1.2

Cancel the common factors.

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Step 49.1.1.2.1

Factor $2$ out of $8$ .

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

Step 49.1.1.2.2

Cancel the common factor.

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

Step 49.1.1.2.3

Rewrite the expression.

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

Step 49.1.2

Multiply $2$ by $2$ .

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

Step 49.2

Apply the distributive property.

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

Step 49.3

Simplify.

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Step 49.3.1

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

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Step 49.3.1.1

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

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

Step 49.3.1.2

Multiply $2$ by $4$ .

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

Step 49.3.2

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

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Step 49.3.2.1

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

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

Step 49.3.2.2

Multiply $2$ by $16$ .

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

Step 49.3.3

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

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Step 49.3.3.1

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

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

Step 49.3.3.2

Multiply $2$ by $2$ .

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

Step 49.4

Simplify each term.

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Step 49.4.1

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

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Step 49.4.1.1

Factor $2$ out of $2 x$ .

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

Step 49.4.1.2

Cancel the common factors.

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Step 49.4.1.2.1

Factor $2$ out of $4$ .

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

Step 49.4.1.2.2

Cancel the common factor.

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

Step 49.4.1.2.3

Rewrite the expression.

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

Step 49.4.2

Multiply $2$ by $2$ .

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

Step 49.4.3

Simplify the numerator.

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Step 49.4.3.1

Apply the distributive property.

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

Step 49.4.3.2

Cancel the common factor of $2$ .

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Step 49.4.3.2.1

Factor $2$ out of $4$ .

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

Step 49.4.3.2.2

Factor $2$ out of $2 x$ .

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

Step 49.4.3.2.3

Cancel the common factor.

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

Step 49.4.3.2.4

Rewrite the expression.

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

Step 49.4.3.3

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

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

Step 49.4.3.4

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

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Step 49.4.3.4.1

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

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

Step 49.4.3.4.2

Multiply $4$ by $2$ .

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

Step 49.5

Combine the numerators over the common denominator.

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

Step 49.6

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

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

Step 49.7

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

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

Step 49.8

Combine the numerators over the common denominator.

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

Step 49.9

Add $x \cdot 2$ and $x$ .

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Step 49.9.1

Reorder $x$ and $2$ .

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

Step 49.9.2

Add $2 \cdot x$ and $x$ .

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

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

Step 49.10

Multiply $- \frac{1}{2} \cdot \frac{\frac{3 x}{2} + \frac{\sin (4 x)}{8} - \sin (2 x)}{2}$ .

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Step 49.10.1

Multiply $\frac{\frac{3 x}{2} + \frac{\sin (4 x)}{8} - \sin (2 x)}{2}$ by $\frac{1}{2}$ .

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

Step 49.10.2

Multiply $2$ by $2$ .

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

Step 50

Reorder terms.

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

Step 51

The answer is the antiderivative of the function $f (x) = \cos^{4} (x) - \sin^{4} (x)$ .

$F (x) =$ $\frac{3}{8} x + \frac{1}{32} \sin (4 x) + \frac{1}{4} \sin (2 x) - \frac{1}{4} (\frac{3}{2} x + \frac{1}{8} \sin (4 x) - \sin (2 x)) + C$