Calculus Examples

$\int_{0}^{π} 2 \sin^{4} (x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{0}^{π} \sin^{4} (x) d x$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $2$ out of $4$ .

$2 \int_{0}^{π} {\sin (x)}^{2 (2)} d x$

Step 2.2

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

$2 \int_{0}^{π} {(\sin^{2} (x))}^{2} d x$

$2 \int_{0}^{π} {(\sin^{2} (x))}^{2} d x$

Step 3

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

$2 \int_{0}^{π} {(\frac{1 - \cos (2 x)}{2})}^{2} d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $2 x$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $2$ by $1$ .

$2$

$2$

Step 4.2

Substitute the lower limit in for $x$ in $u_{1} = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 4.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $x$ in $u_{1} = 2 x$ .

$u_{upper} = 2 π$

Step 4.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 4.6

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$2 \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1}{2} d u_{1}$

$2 \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} \frac{1}{2} d u_{1}$

Step 5

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

$2 (\frac{1}{2} \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1})$

Step 6

Simplify by multiplying through.

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Step 6.1

Simplify.

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Step 6.1.1

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

$\frac{2}{2} \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Cancel the common factor.

$\frac{2}{2} \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$

Step 6.1.2.2

Rewrite the expression.

$1 \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$

$1 \int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$

Step 6.1.3

Multiply $\int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$ by $1$ .

$\int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$

$\int_{0}^{2 π} {(\frac{1 - \cos (u_{1})}{2})}^{2} d u_{1}$

Step 6.2

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

$\int_{0}^{2 π} {(\frac{1}{2} \cdot (1 - \cos (u_{1})))}^{2} d u_{1}$

Step 6.3

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

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Step 6.3.1

Rewrite the exponentiation as a product.

$\int_{0}^{2 π} \frac{1}{2} \cdot (1 - \cos (u_{1})) (\frac{1}{2} \cdot (1 - \cos (u_{1}))) d u_{1}$

Step 6.3.2

Apply the distributive property.

$\int_{0}^{2 π} (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{2} \cdot (1 - \cos (u_{1}))) d u_{1}$

Step 6.3.3

Apply the distributive property.

$\int_{0}^{2 π} (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.4

Apply the distributive property.

$\int_{0}^{2 π} \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.5

Apply the distributive property.

$\int_{0}^{2 π} \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.6

Apply the distributive property.

$\int_{0}^{2 π} \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.7

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

$\int_{0}^{2 π} 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.8

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

$\int_{0}^{2 π} 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.9

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.10

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot (- \cos (u_{1}))) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.11

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2} \cos (u_{1})) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.12

Move parentheses.

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (- 1 \cdot \frac{1}{2}) \cos (u_{1}) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.13

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.14

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.15

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot \frac{1}{2} \cos (u_{1}) (1 \cdot \frac{1}{2}) + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.16

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot \frac{1}{2} \cdot 1 \cos (u_{1}) \cdot \frac{1}{2} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.17

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} + \frac{1}{2} \cdot (- \cos (u_{1})) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.18

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cos (u_{1}) (\frac{1}{2} \cdot (- \cos (u_{1}))) d u_{1}$

Step 6.3.19

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cos (u_{1}) (- 1 \cdot \frac{1}{2} \cos (u_{1})) d u_{1}$

Step 6.3.20

Move parentheses.

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cos (u_{1}) (- 1 \cdot \frac{1}{2}) \cos (u_{1}) d u_{1}$

Step 6.3.21

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} - 1 \cdot \frac{1}{2} \cdot - 1 \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.22

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

$\int_{0}^{2 π} 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.23

Multiply $1$ by $1$ .

$\int_{0}^{2 π} 1 (\frac{1}{2}) \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.24

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

$\int_{0}^{2 π} \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.25

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

$\int_{0}^{2 π} \frac{1}{2 \cdot 2} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.26

Multiply $2$ by $2$ .

$\int_{0}^{2 π} \frac{1}{4} + 1 \cdot - 1 \frac{1}{2} \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.27

Multiply $- 1$ by $1$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.28

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{1}{2 \cdot 2} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.29

Multiply $2$ by $2$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{1}{4} \cos (u_{1}) - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.30

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - 1 \cdot 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.31

Multiply $- 1$ by $1$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{1}{2} \cos (u_{1}) \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.32

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.33

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{2 \cdot 2} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.34

Multiply $2$ by $2$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} - 1 \cdot - 1 \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.35

Multiply $- 1$ by $- 1$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.36

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.37

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 6.3.38

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{2 \cdot 2} \cos (u_{1}) d u_{1}$

Step 6.3.39

Multiply $2$ by $2$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} \cos (u_{1}) d u_{1}$

Step 6.3.40

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1}) \cos (u_{1})}{4} d u_{1}$

Step 6.3.41

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{4} d u_{1}$

Step 6.3.42

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{4} d u_{1}$

Step 6.3.43

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

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{{\cos (u_{1})}^{1 + 1}}{4} d u_{1}$

Step 6.3.44

Add $1$ and $1$ .

$\int_{0}^{2 π} \frac{1}{4} - \frac{\cos (u_{1})}{4} - \frac{\cos (u_{1})}{4} + \frac{\cos^{2} (u_{1})}{4} d u_{1}$

Step 6.3.45

Subtract $\frac{\cos (u_{1})}{4}$ from $- \frac{\cos (u_{1})}{4}$ .

$\int_{0}^{2 π} \frac{1}{4} - 2 \frac{\cos (u_{1})}{4} + \frac{\cos^{2} (u_{1})}{4} d u_{1}$

Step 6.3.46

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

$\int_{0}^{2 π} \frac{1}{4} + \frac{- 2 \cos (u_{1})}{4} + \frac{\cos^{2} (u_{1})}{4} d u_{1}$

Step 6.3.47

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

$\int_{0}^{2 π} \frac{1}{4} + \frac{\cos^{2} (u_{1})}{4} + \frac{- 2 \cos (u_{1})}{4} d u_{1}$

Step 6.3.48

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

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{- 2 \cos (u_{1})}{4} d u_{1}$

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{- 2 \cos (u_{1})}{4} d u_{1}$

Step 6.4

Simplify.

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Step 6.4.1

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

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Step 6.4.1.1

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

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 (- \cos (u_{1}))}{4} d u_{1}$

Step 6.4.1.2

Cancel the common factors.

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Step 6.4.1.2.1

Factor $2$ out of $4$ .

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 (- \cos (u_{1}))}{2 (2)} d u_{1}$

Step 6.4.1.2.2

Cancel the common factor.

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 (- \cos (u_{1}))}{2 \cdot 2} d u_{1}$

Step 6.4.1.2.3

Rewrite the expression.

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{- \cos (u_{1})}{2} d u_{1}$

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{- \cos (u_{1})}{2} d u_{1}$

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{- \cos (u_{1})}{2} d u_{1}$

Step 6.4.2

Move the negative in front of the fraction.

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} - \frac{\cos (u_{1})}{2} d u_{1}$

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} - \frac{\cos (u_{1})}{2} d u_{1}$

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 7

Split the single integral into multiple integrals.

$\int_{0}^{2 π} \frac{\cos^{2} (u_{1})}{4} d u_{1} + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 8

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

$\frac{1}{4} \int_{0}^{2 π} \cos^{2} (u_{1}) d u_{1} + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 9

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

$\frac{1}{4} \int_{0}^{2 π} \frac{1 + \cos (2 u_{1})}{2} d u_{1} + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 10

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

$\frac{1}{4} (\frac{1}{2} \int_{0}^{2 π} 1 + \cos (2 u_{1}) d u_{1}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 11

Simplify.

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Step 11.1

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

$\frac{1}{2 \cdot 4} \int_{0}^{2 π} 1 + \cos (2 u_{1}) d u_{1} + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 11.2

Multiply $2$ by $4$ .

$\frac{1}{8} \int_{0}^{2 π} 1 + \cos (2 u_{1}) d u_{1} + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

$\frac{1}{8} \int_{0}^{2 π} 1 + \cos (2 u_{1}) d u_{1} + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 12

Split the single integral into multiple integrals.

$\frac{1}{8} (\int_{0}^{2 π} d u_{1} + \int_{0}^{2 π} \cos (2 u_{1}) d u_{1}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 13

Apply the constant rule.

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \int_{0}^{2 π} \cos (2 u_{1}) d u_{1}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 14

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 14.1

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

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Step 14.1.1

Differentiate $2 u_{1}$ .

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

Step 14.1.2

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 14.1.3

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 14.1.4

Multiply $2$ by $1$ .

$2$

$2$

Step 14.2

Substitute the lower limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{lower} = 2 \cdot 0$

Step 14.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 14.4

Substitute the upper limit in for $u_{1}$ in $u_{2} = 2 u_{1}$ .

$u_{upper} = 2 (2 π)$

Step 14.5

Multiply $2$ by $2$ .

$u_{upper} = 4 π$

Step 14.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 4 π$

Step 14.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \int_{0}^{4 π} \cos (u_{2}) \frac{1}{2} d u_{2}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \int_{0}^{4 π} \cos (u_{2}) \frac{1}{2} d u_{2}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 15

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

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \int_{0}^{4 π} \frac{\cos (u_{2})}{2} d u_{2}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 16

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

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \frac{1}{2} \int_{0}^{4 π} \cos (u_{2}) d u_{2}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 17

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

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \frac{1}{2} \sin (u_{2})]_{0}^{4 π}) + \int_{0}^{2 π} \frac{1}{4} d u_{1} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 18

Apply the constant rule.

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \frac{1}{2} \sin (u_{2})]_{0}^{4 π}) + \frac{1}{4} u_{1}]_{0}^{2 π} + \int_{0}^{2 π} - \frac{\cos (u_{1})}{2} d u_{1}$

Step 19

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

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \frac{1}{2} \sin (u_{2})]_{0}^{4 π}) + \frac{1}{4} u_{1}]_{0}^{2 π} - \int_{0}^{2 π} \frac{\cos (u_{1})}{2} d u_{1}$

Step 20

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

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \frac{1}{2} \sin (u_{2})]_{0}^{4 π}) + \frac{1}{4} u_{1}]_{0}^{2 π} - (\frac{1}{2} \int_{0}^{2 π} \cos (u_{1}) d u_{1})$

Step 21

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

$\frac{1}{8} (u_{1}]_{0}^{2 π} + \frac{1}{2} \sin (u_{2})]_{0}^{4 π}) + \frac{1}{4} u_{1}]_{0}^{2 π} - \frac{1}{2} \sin (u_{1})]_{0}^{2 π}$

Step 22

Substitute and simplify.

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Step 22.1

Evaluate $u_{1}$ at $2 π$ and at $0$ .

$\frac{1}{8} ((2 π) + 0 + \frac{1}{2} \sin (u_{2})]_{0}^{4 π}) + \frac{1}{4} u_{1}]_{0}^{2 π} - \frac{1}{2} \sin (u_{1})]_{0}^{2 π}$

Step 22.2

Evaluate $\sin (u_{2})$ at $4 π$ and at $0$ .

$\frac{1}{8} ((2 π) + 0 + \frac{1}{2} ((\sin (4 π)) - \sin (0))) + \frac{1}{4} u_{1}]_{0}^{2 π} - \frac{1}{2} \sin (u_{1})]_{0}^{2 π}$

Step 22.3

Evaluate $\frac{1}{4} u_{1}$ at $2 π$ and at $0$ .

$\frac{1}{8} ((2 π) + 0 + \frac{1}{2} ((\sin (4 π)) - \sin (0))) + (\frac{1}{4} (2 π)) - \frac{1}{4} \cdot 0 - \frac{1}{2} \sin (u_{1})]_{0}^{2 π}$

Step 22.4

Evaluate $\sin (u_{1})$ at $2 π$ and at $0$ .

$\frac{1}{8} ((2 π) + 0 + \frac{1}{2} ((\sin (4 π)) - \sin (0))) + (\frac{1}{4} (2 π)) - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5

Simplify.

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Step 22.5.1

Add $2 π$ and $0$ .

$\frac{1}{8} (2 π + \frac{1}{2} ((\sin (4 π)) - \sin (0))) + (\frac{1}{4} (2 π)) - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.2

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

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{2}{4} π - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.3

Combine $\frac{2}{4}$ and $π$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{2 π}{4} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.4

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

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Step 22.5.4.1

Factor $2$ out of $2 π$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{2 (π)}{4} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.4.2

Cancel the common factors.

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Step 22.5.4.2.1

Factor $2$ out of $4$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{2 π}{2 \cdot 2} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.4.2.2

Cancel the common factor.

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{2 π}{2 \cdot 2} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.4.2.3

Rewrite the expression.

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} - \frac{1}{4} \cdot 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.5

Multiply $0$ by $- 1$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} + 0 (\frac{1}{4}) - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.6

Multiply $0$ by $\frac{1}{4}$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} + 0 - \frac{1}{2} ((\sin (2 π)) - \sin (0))$

Step 22.5.7

Add $\frac{π}{2}$ and $0$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - \sin (0))$

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - \sin (0))$

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - \sin (0))) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - \sin (0))$

Step 23

Simplify.

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Step 23.1

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - 0)) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - \sin (0))$

Step 23.2

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) - 0)) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - 0)$

Step 23.3

Multiply $- 1$ by $0$ .

$\frac{1}{8} (2 π + \frac{1}{2} (\sin (4 π) + 0)) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - 0)$

Step 23.4

Add $\sin (4 π)$ and $0$ .

$\frac{1}{8} (2 π + \frac{1}{2} \sin (4 π)) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - 0)$

Step 23.5

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

$\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) - 0)$

Step 23.6

Multiply $- 1$ by $0$ .

$\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) + \frac{π}{2} - \frac{1}{2} (\sin (2 π) + 0)$

Step 23.7

Add $\sin (2 π)$ and $0$ .

$\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) + \frac{π}{2} - \frac{1}{2} \sin (2 π)$

Step 23.8

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

$\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) + \frac{π}{2} - \frac{\sin (2 π)}{2}$

Step 23.9

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

$\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) \cdot \frac{2}{2} - \frac{\sin (2 π)}{2} + \frac{π}{2}$

Step 23.10

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

$\frac{\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) \cdot 2}{2} - \frac{\sin (2 π)}{2} + \frac{π}{2}$

Step 23.11

Combine the numerators over the common denominator.

$\frac{\frac{1}{8} (2 π + \frac{\sin (4 π)}{2}) \cdot 2 - \sin (2 π)}{2} + \frac{π}{2}$

Step 23.12

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

$\frac{\frac{2}{8} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 23.13

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

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Step 23.13.1

Factor $2$ out of $2$ .

$\frac{\frac{2 (1)}{8} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 23.13.2

Cancel the common factors.

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Step 23.13.2.1

Factor $2$ out of $8$ .

$\frac{\frac{2 \cdot 1}{2 \cdot 4} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 23.13.2.2

Cancel the common factor.

$\frac{\frac{2 \cdot 1}{2 \cdot 4} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 23.13.2.3

Rewrite the expression.

$\frac{\frac{1}{4} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

$\frac{\frac{1}{4} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

$\frac{\frac{1}{4} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

$\frac{\frac{1}{4} (2 π + \frac{\sin (4 π)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24

Simplify.

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Step 24.1

Simplify each term.

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Step 24.1.1

Simplify the numerator.

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Step 24.1.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\frac{1}{4} (2 π + \frac{\sin (0)}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.1.1.2

The exact value of $\sin (0)$ is $0$ .

$\frac{\frac{1}{4} (2 π + \frac{0}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

$\frac{\frac{1}{4} (2 π + \frac{0}{2}) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.1.2

Divide $0$ by $2$ .

$\frac{\frac{1}{4} (2 π + 0) - \sin (2 π)}{2} + \frac{π}{2}$

$\frac{\frac{1}{4} (2 π + 0) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.2

Add $2 π$ and $0$ .

$\frac{\frac{1}{4} (2 π) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.3

Cancel the common factor of $2$ .

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Step 24.3.1

Factor $2$ out of $4$ .

$\frac{\frac{1}{2 (2)} (2 π) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.3.2

Factor $2$ out of $2 π$ .

$\frac{\frac{1}{2 (2)} (2 (π)) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.3.3

Cancel the common factor.

$\frac{\frac{1}{2 \cdot 2} (2 π) - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.3.4

Rewrite the expression.

$\frac{\frac{1}{2} π - \sin (2 π)}{2} + \frac{π}{2}$

$\frac{\frac{1}{2} π - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.4

Combine $\frac{1}{2}$ and $π$ .

$\frac{\frac{π}{2} - \sin (2 π)}{2} + \frac{π}{2}$

Step 24.5

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\frac{π}{2} - \sin (0)}{2} + \frac{π}{2}$

Step 24.6

The exact value of $\sin (0)$ is $0$ .

$\frac{\frac{π}{2} - 0}{2} + \frac{π}{2}$

Step 24.7

Multiply $- 1$ by $0$ .

$\frac{\frac{π}{2} + 0}{2} + \frac{π}{2}$

Step 24.8

Add $\frac{π}{2}$ and $0$ .

$\frac{\frac{π}{2}}{2} + \frac{π}{2}$

Step 24.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 24.10

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

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Step 24.10.1

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

$\frac{π}{2 \cdot 2} + \frac{π}{2}$

Step 24.10.2

Multiply $2$ by $2$ .

$\frac{π}{4} + \frac{π}{2}$

$\frac{π}{4} + \frac{π}{2}$

Step 24.11

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

$\frac{π}{4} + \frac{π}{2} \cdot \frac{2}{2}$

Step 24.12

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 24.12.1

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

$\frac{π}{4} + \frac{π \cdot 2}{2 \cdot 2}$

Step 24.12.2

Multiply $2$ by $2$ .

$\frac{π}{4} + \frac{π \cdot 2}{4}$

$\frac{π}{4} + \frac{π \cdot 2}{4}$

Step 24.13

Combine the numerators over the common denominator.

$\frac{π + π \cdot 2}{4}$

Step 24.14

Add $π$ and $π \cdot 2$ .

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Step 24.14.1

Reorder $π$ and $2$ .

$\frac{π + 2 \cdot π}{4}$

Step 24.14.2

Add $π$ and $2 \cdot π$ .

$\frac{3 π}{4}$

$\frac{3 π}{4}$

$\frac{3 π}{4}$

Step 25

The result can be shown in multiple forms.

Exact Form:

$\frac{3 π}{4}$

Decimal Form:

$2.35619449 \dots$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$