Calculus Examples

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

Step 1

Write $\cos^{2} (\frac{x}{2})$ as a function.

$f (x) = \cos^{2} (\frac{x}{2})$

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.

$F (x) = \int \cos^{2} (\frac{x}{2}) d x$

Step 4

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

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

$\frac{1}{2}$

Step 4.2

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

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

Step 9.3

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

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

Apply the constant rule.

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

Step 12

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 12.1

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

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Step 12.1.1

Differentiate $2 u_{1}$ .

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

Step 12.1.2

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 12.1.3

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 12.1.4

Multiply $2$ by $1$ .

$2$

Step 12.2

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

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

$\frac{x}{2} + \frac{1}{2} \sin (u_{2}) + C$

Step 17.2

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

$\frac{x}{2} + \frac{1}{2} \sin (2 u_{1}) + C$

Step 17.3

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

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

Step 18

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

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

Step 19

Simplify.

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Step 19.1

Cancel the common factor of $2$ .

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Step 19.1.1

Cancel the common factor.

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

Step 19.1.2

Divide $x$ by $1$ .

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

Step 19.2

Reorder terms.

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

Step 20

The answer is the antiderivative of the function $f (x) = \cos^{2} (\frac{x}{2})$ .

$F (x) =$ $\frac{1}{2} x + \frac{1}{2} \sin (x) + C$