Calculus Examples

$\arctan (\frac{x}{2})$

Step 1

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

$f (x) = \arctan (\frac{x}{2})$

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.

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

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arctan (\frac{x}{2})$ and $d v = 1$ .

$\arctan (\frac{x}{2}) x - \int x \frac{2}{x^{2} + 4} d x$

Step 5

Simplify.

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Step 5.1

Combine $x$ and $\frac{2}{x^{2} + 4}$ .

$\arctan (\frac{x}{2}) x - \int \frac{x \cdot 2}{x^{2} + 4} d x$

Step 5.2

Move $2$ to the left of $x$ .

$\arctan (\frac{x}{2}) x - \int \frac{2 x}{x^{2} + 4} d x$

Step 6

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

$\arctan (\frac{x}{2}) x - (2 \int \frac{x}{x^{2} + 4} d x)$

Step 7

Multiply $2$ by $- 1$ .

$\arctan (\frac{x}{2}) x - 2 \int \frac{x}{x^{2} + 4} d x$

Step 8

Let $u = x^{2} + 4$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $x^{2} + 4$ .

$\frac{d}{d x} [x^{2} + 4]$

Step 8.1.2

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 8.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [4]$

Step 8.1.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$2 x + 0$

Step 8.1.5

Add $2 x$ and $0$ .

$2 x$

Step 8.2

Rewrite the problem using $u$ and $d u$ .

$\arctan (\frac{x}{2}) x - 2 \int \frac{1}{u} \cdot \frac{1}{2} d u$

Step 9

Simplify.

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Step 9.1

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

$\arctan (\frac{x}{2}) x - 2 \int \frac{1}{u \cdot 2} d u$

Step 9.2

Move $2$ to the left of $u$ .

$\arctan (\frac{x}{2}) x - 2 \int \frac{1}{2 u} d u$

Step 10

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

$\arctan (\frac{x}{2}) x - 2 (\frac{1}{2} \int \frac{1}{u} d u)$

Step 11

Simplify.

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Step 11.1

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

$\arctan (\frac{x}{2}) x + \frac{- 2}{2} \int \frac{1}{u} d u$

Step 11.2

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

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Step 11.2.1

Factor $2$ out of $- 2$ .

$\arctan (\frac{x}{2}) x + \frac{2 \cdot - 1}{2} \int \frac{1}{u} d u$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $2$ out of $2$ .

$\arctan (\frac{x}{2}) x + \frac{2 \cdot - 1}{2 (1)} \int \frac{1}{u} d u$

Step 11.2.2.2

Cancel the common factor.

$\arctan (\frac{x}{2}) x + \frac{2 \cdot - 1}{2 \cdot 1} \int \frac{1}{u} d u$

Step 11.2.2.3

Rewrite the expression.

$\arctan (\frac{x}{2}) x + \frac{- 1}{1} \int \frac{1}{u} d u$

Step 11.2.2.4

Divide $- 1$ by $1$ .

$\arctan (\frac{x}{2}) x - \int \frac{1}{u} d u$

Step 12

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\arctan (\frac{x}{2}) x - (\ln (| u |) + C)$

Step 13

Rewrite $\arctan (\frac{x}{2}) x - (\ln (| u |) + C)$ as $\arctan (\frac{1}{2} x) x - \ln (| u |) + C$ .

$\arctan (\frac{1}{2} x) x - \ln (| u |) + C$

Step 14

Replace all occurrences of $u$ with $x^{2} + 4$ .

$\arctan (\frac{1}{2} x) x - \ln (| x^{2} + 4 |) + C$

Step 15

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

$F (x) =$ $\arctan (\frac{1}{2} x) x - \ln (| x^{2} + 4 |) + C$