Calculus Examples

$\int x {(x^{2} - 1)}^{8} d x$

Step 1

Let $u = x^{2} - 1$ . 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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Let $u = x^{2} - 1$ . Find $\frac{d u}{d x}$ .

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Differentiate $x^{2} - 1$ .

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

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

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

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

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$2 x + 0$

Add $2 x$ and $0$ .

$2 x$

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

$\int u^{8} \frac{1}{2} d u$

Step 2

Combine $u^{8}$ and $\frac{1}{2}$ .

$\int \frac{u^{8}}{2} d u$

Step 3

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

$\frac{1}{2} \int u^{8} d u$

Step 4

By the Power Rule, the integral of $u^{8}$ with respect to $u$ is $\frac{1}{9} u^{9}$ .

$\frac{1}{2} (\frac{1}{9} u^{9} + C)$

Step 5

Simplify.

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Rewrite $\frac{1}{2} (\frac{1}{9} u^{9} + C)$ as $\frac{1}{2} \cdot \frac{1}{9} u^{9} + C$ .

$\frac{1}{2} \cdot \frac{1}{9} u^{9} + C$

Simplify.

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Multiply $\frac{1}{2}$ by $\frac{1}{9}$ .

$\frac{1}{2 \cdot 9} u^{9} + C$

Multiply $2$ by $9$ .

$\frac{1}{18} u^{9} + C$

Step 6

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

$\frac{1}{18} {(x^{2} - 1)}^{9} + C$

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