Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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 5.1.3

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 5.1.4

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

$\frac{1}{2}$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

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

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

Step 7

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

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

Step 8

Multiply $2$ by $2$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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Step 11.2.1

Factor $2$ out of $4$ .

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

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $2$ out of $2$ .

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

Step 11.2.2.2

Cancel the common factor.

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

Step 11.2.2.3

Rewrite the expression.

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

Step 11.2.2.4

Divide $2$ by $1$ .

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

Step 12

Split the single integral into multiple integrals.

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

Step 13

Apply the constant rule.

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

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 $1$ .

$2$

Step 14.2

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

Simplify.

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

Step 19

Substitute back in for each integration substitution variable.

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Step 19.1

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

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

Step 19.2

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

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

Step 19.3

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

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

Step 20

Simplify.

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Step 20.1

Simplify each term.

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Step 20.1.1

Cancel the common factor of $2$ .

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Step 20.1.1.1

Cancel the common factor.

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

Step 20.1.1.2

Rewrite the expression.

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

Step 20.1.2

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

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

Step 20.2

Apply the distributive property.

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

Step 20.3

Cancel the common factor of $2$ .

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Step 20.3.1

Cancel the common factor.

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

Step 20.3.2

Rewrite the expression.

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

Step 20.4

Cancel the common factor of $2$ .

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Step 20.4.1

Cancel the common factor.

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

Step 20.4.2

Rewrite the expression.

$x + \sin (x) + C$

Step 21

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

$F (x) =$ $x + \sin (x) + C$