Calculus Examples

$2 \arctan (x)$

Step 1

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

$f (x) = 2 \arctan (x)$

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.

$F (x) = \int 2 \arctan (x) d x$

Step 4

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

$2 \int \arctan (x) d x$

Step 5

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

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

Step 6

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

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

Step 7

Let $u = x^{2} + 1$ . 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 7.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 7.1.1

Differentiate $x^{2} + 1$ .

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

Step 7.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

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

Step 7.1.3

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

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

Step 7.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 7.1.5

Add $2 x$ and $0$ .

$2 x$

Step 7.2

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

Replace all occurrences of $u$ with $x^{2} + 1$ .

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

Step 13

Simplify.

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Step 13.1

Combine $\ln (| x^{2} + 1 |)$ and $\frac{1}{2}$ .

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

Step 13.2

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

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

Step 13.3

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

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

Step 13.4

Combine the numerators over the common denominator.

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

Step 13.5

Cancel the common factor of $2$ .

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Step 13.5.1

Cancel the common factor.

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

Step 13.5.2

Rewrite the expression.

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

Step 13.6

Move $2$ to the left of $\arctan (x) x$ .

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

Step 13.7

Reorder factors in $2 \arctan (x) x - \ln (| x^{2} + 1 |)$ .

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

Step 14

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

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