Calculus Examples

$\int \frac{9 x {(x^{2} - 10)}^{8}}{5} d x$

Step 1

Since $\frac{9}{5}$ is constant with respect to $x$ , move $\frac{9}{5}$ out of the integral.

$\frac{9}{5} \int x {(x^{2} - 10)}^{8} d x$

Step 2

Let $u = x^{2} - 10$ . 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 2.1

Let $u = x^{2} - 10$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{2} - 10$ .

$\frac{d}{d x} [x^{2} - 10]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{2} - 10$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10]$

Step 2.1.3

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

$2 x + \frac{d}{d x} [- 10]$

Step 2.1.4

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

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

$\frac{9}{5} \int u^{8} \frac{1}{2} d u$

Step 3

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

$\frac{9}{5} \int \frac{u^{8}}{2} d u$

Step 4

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

$\frac{9}{5} (\frac{1}{2} \int u^{8} d u)$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{2}$ by $\frac{9}{5}$ .

$\frac{9}{2 \cdot 5} \int u^{8} d u$

Step 5.2

Multiply $2$ by $5$ .

$\frac{9}{10} \int u^{8} d u$

Step 6

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

$\frac{9}{10} (\frac{1}{9} u^{9} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{9}{10} (\frac{1}{9} u^{9} + C)$ as $\frac{9}{10} \cdot \frac{1}{9} u^{9} + C$ .

$\frac{9}{10} \cdot \frac{1}{9} u^{9} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{9}{10}$ by $\frac{1}{9}$ .

$\frac{9}{10 \cdot 9} u^{9} + C$

Step 7.2.2

Multiply $10$ by $9$ .

$\frac{9}{90} u^{9} + C$

Step 7.2.3

Cancel the common factor of $9$ and $90$ .

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Step 7.2.3.1

Factor $9$ out of $9$ .

$\frac{9 (1)}{90} u^{9} + C$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $9$ out of $90$ .

$\frac{9 \cdot 1}{9 \cdot 10} u^{9} + C$

Step 7.2.3.2.2

Cancel the common factor.

$\frac{9 \cdot 1}{9 \cdot 10} u^{9} + C$

Step 7.2.3.2.3

Rewrite the expression.

$\frac{1}{10} u^{9} + C$

Step 8

Replace all occurrences of $u$ with $x^{2} - 10$ .

$\frac{1}{10} {(x^{2} - 10)}^{9} + C$