Calculus Examples

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

Step 1

Split the fraction $\frac{1 + x}{1 + x^{2}}$ into two fractions.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Rewrite $1$ as $1^{2}$ .

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

Step 4

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C$ .

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

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 $d u$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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

Step 10

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

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