Calculus Examples

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

Step 1

Write $\frac{2 x}{1 + x^{2}}$ as a function.

$f (x) = \frac{2 x}{1 + x^{2}}$

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.

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

Step 4

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

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

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 $d u$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Cancel the common factor.

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

Step 8.2.2

Rewrite the expression.

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

Step 8.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

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

Step 9

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

$\ln (| u |) + C$

Step 10

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

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

Step 11

The answer is the antiderivative of the function $f (x) = \frac{2 x}{1 + x^{2}}$ .

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