Calculus Examples

$\int {(x^{2} - 9)}^{3} (2 x) d x$

Step 1

Move $2$ to the left of ${(x^{2} - 9)}^{3}$ .

$\int 2 {(x^{2} - 9)}^{3} x d x$

Step 2

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

$2 \int {(x^{2} - 9)}^{3} x d x$

Step 3

Let $u = x^{2} - 9$ . 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 3.1

Let $u = x^{2} - 9$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $x^{2} - 9$ .

$\frac{d}{d x} [x^{2} - 9]$

Step 3.1.2

By the Sum Rule, the derivative of $x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]$

Step 3.1.3

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

$2 x + \frac{d}{d x} [- 9]$

Step 3.1.4

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$2 x + 0$

Step 3.1.5

Add $2 x$ and $0$ .

$2 x$

Step 3.2

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

$2 \int u^{3} \frac{1}{2} d u$

Step 4

Combine $u^{3}$ and $\frac{1}{2}$ .

$2 \int \frac{u^{3}}{2} d u$

Step 5

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

$2 (\frac{1}{2} \int u^{3} d u)$

Step 6

Simplify.

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Step 6.1

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

$\frac{2}{2} \int u^{3} d u$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Cancel the common factor.

$\frac{2}{2} \int u^{3} d u$

Step 6.2.2

Rewrite the expression.

$1 \int u^{3} d u$

Step 6.3

Multiply $\int u^{3} d u$ by $1$ .

$\int u^{3} d u$

Step 7

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{4} u^{4} + C$

Step 8

Replace all occurrences of $u$ with $x^{2} - 9$ .

$\frac{1}{4} {(x^{2} - 9)}^{4} + C$