Calculus Examples

$- \frac{2 x}{x^{2} - 8}$

Step 1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int \frac{2 x}{x^{2} - 8} d x$

Step 2

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

$- (2 \int \frac{x}{x^{2} - 8} d x)$

Step 3

Multiply $2$ by $- 1$ .

$- 2 \int \frac{x}{x^{2} - 8} d x$

Step 4

Let $u = x^{2} - 8$ . 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 4.1

Let $u = x^{2} - 8$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $x^{2} - 8$ .

$\frac{d}{d x} [x^{2} - 8]$

Step 4.1.2

By the Sum Rule, the derivative of $x^{2} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 8]$

Step 4.1.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$2 x + 0$

Step 4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 4.2

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 6

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 7

Simplify.

Tap for more steps...

Step 7.1

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

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

Step 7.2

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 7.2.1

Factor $2$ out of $- 2$ .

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

Step 7.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.2.1

Factor $2$ out of $2$ .

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

Step 7.2.2.2

Cancel the common factor.

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

Step 7.2.2.3

Rewrite the expression.

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

Step 7.2.2.4

Divide $- 1$ by $1$ .

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

Step 8

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

$- (\ln (| u |) + C)$

Step 9

Simplify.

$- \ln (| u |) + C$

Step 10

Replace all occurrences of $u$ with $x^{2} - 8$ .

$- \ln (| x^{2} - 8 |) + C$