Calculus Examples

\int \arctan (x) d x

$\int \arctan (x) d x$

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = \arctan (x)

$u = \arctan (x)$ and

d v = 1

$d v = 1$ .

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

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

Step 2

Combine

x

$x$ and

\frac{1}{x^{2} + 1}

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

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

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

Step 3

Let

u = x^{2} + 1

$u = x^{2} + 1$ . Then

d u = 2 x d x

$d u = 2 x d x$ , so

\frac{1}{2} d u = x d x

$\frac{1}{2} d u = x d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 3.1

Let

u = x^{2} + 1

$u = x^{2} + 1$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate

x^{2} + 1

$x^{2} + 1$ .

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

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

Step 3.1.2

By the Sum Rule, the derivative of

x^{2} + 1

$x^{2} + 1$ with respect to

x

$x$ is

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

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

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

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

Step 3.1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

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

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

Step 3.1.4

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

2 x + 0

$2 x + 0$

Step 3.1.5

Add

2 x

$2 x$ and

0

$0$ .

2 x

$2 x$

2 x

$2 x$

Step 3.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 4

Simplify.

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Step 4.1

Multiply

\frac{1}{u}

$\frac{1}{u}$ by

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 4.2

Move

2

$2$ to the left of

u

$u$ .

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

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

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

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

Step 5

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

u

$u$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

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

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

Step 6

The integral of

\frac{1}{u}

$\frac{1}{u}$ with respect to

u

$u$ is

\ln (| u |)

$\ln (| u |)$ .

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

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

Step 7

Simplify.

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

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

Step 8

Replace all occurrences of

u

$u$ with

x^{2} + 1

$x^{2} + 1$ .

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

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