Calculus Examples

$\int x {(x^{2} - 1)}^{4} d x$

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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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

$\int u^{4} \frac{1}{2} d u$

Step 2

Combine $u^{4}$ and $\frac{1}{2}$ .

$\int \frac{u^{4}}{2} d u$

Step 3

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

$\frac{1}{2} \int u^{4} d u$

Step 4

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{2} (\frac{1}{5} u^{5} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{2} (\frac{1}{5} u^{5} + C)$ as $\frac{1}{2} \cdot \frac{1}{5} u^{5} + C$ .

$\frac{1}{2} \cdot \frac{1}{5} u^{5} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{5}$ .

$\frac{1}{2 \cdot 5} u^{5} + C$

Step 5.2.2

Multiply $2$ by $5$ .

$\frac{1}{10} u^{5} + C$

Step 6

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

$\frac{1}{10} {(x^{2} - 1)}^{5} + C$

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