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Calculus Examples

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

$u = x^{2} + 1$

Step 1

Let

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

$d u = 2 x d x$ . Rewrite using

$u$ and

$d u$ .

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Step 1.1

Let

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

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

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Step 1.1.1

Differentiate

$x^{2} + 1$ .

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

Step 1.1.2

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 1.1.3

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 1.1.4

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$2 x + 0$

Step 1.1.5

Add

$2 x$ and

$0$ .

$2 x$

$2 x$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

$\int \frac{1}{2 u} d u$

$\int \frac{1}{2 u} d u$

Step 2

Since

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

$u$ , move

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

$\frac{1}{2} \int \frac{1}{u} d u$

Step 3

The integral of

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

$u$ is

$\ln (| u |)$ .

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

Step 4

Simplify.

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

Step 5

Replace all occurrences of

$u$ with

$x^{2} + 1$ .

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

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$