Calculus Examples

$\int \frac{x}{4 + x^{2}} d x$

Step 1

Let $u = 4 + x^{2}$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 4 + x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $4 + x^{2}$ .

$\frac{d}{d x} [4 + x^{2}]$

Step 1.1.2

By the Sum Rule, the derivative of $4 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [x^{2}]$

Step 1.1.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 1.1.4

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 1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 3

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 4

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

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

Step 5

Simplify.

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

Step 6

Replace all occurrences of $u$ with $4 + x^{2}$ .

$\frac{1}{2} \ln (| 4 + x^{2} |) + C$