Calculus Examples

$\int arccot (x) d x$

Step 1

Integrate by parts using the formula

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

$u = arccot (x)$ and

$d v = 1$ .

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

Step 2

Combine

$x$ and

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

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

Step 3

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

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

Step 4

Simplify.

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Step 4.1

Multiply

$- 1$ by

$- 1$ .

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

Step 4.2

Multiply

$\int \frac{x}{1 + x^{2}} d x$ by

$1$ .

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

Step 5

Let

$u = 1 + x^{2}$ . 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 5.1

Let

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

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

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Step 5.1.1

Differentiate

$1 + x^{2}$ .

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

Step 5.1.2

By the Sum Rule, the derivative of

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

$x$ is

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

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

Step 5.1.3

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 5.1.4

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 2$ .

$0 + 2 x$

Step 5.1.5

Add

$0$ and

$2 x$ .

$2 x$

Step 5.2

Rewrite the problem using

$u$ and

$d u$ .

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

Step 6

Simplify.

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Step 6.1

Multiply

$\frac{1}{u}$ by

$\frac{1}{2}$ .

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

Step 6.2

Move

$2$ to the left of

$u$ .

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

Step 7

Since

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

$u$ , move

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

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

Step 8

The integral of

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

$u$ is

$\ln (| u |)$ .

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

Step 9

Simplify.

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

Step 10

Replace all occurrences of

$u$ with

$1 + x^{2}$ .

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