Calculus Examples

$\int 3 x {(x^{2} - 3)}^{7} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x {(x^{2} - 3)}^{7} d x$

Step 2

Let $u = x^{2} - 3$ . 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 $u = x^{2} - 3$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{2} - 3$ .

$\frac{d}{d x} [x^{2} - 3]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]$

Step 2.1.3

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

$2 x + \frac{d}{d x} [- 3]$

Step 2.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ 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 $d u$ .

$3 \int u^{7} \frac{1}{2} d u$

Step 3

Combine $u^{7}$ and $\frac{1}{2}$ .

$3 \int \frac{u^{7}}{2} d u$

Step 4

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

$3 (\frac{1}{2} \int u^{7} d u)$

Step 5

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

$\frac{3}{2} \int u^{7} d u$

Step 6

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$\frac{3}{2} (\frac{1}{8} u^{8} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{3}{2} (\frac{1}{8} u^{8} + C)$ as $\frac{3}{2} \cdot \frac{1}{8} u^{8} + C$ .

$\frac{3}{2} \cdot \frac{1}{8} u^{8} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{3}{2}$ by $\frac{1}{8}$ .

$\frac{3}{2 \cdot 8} u^{8} + C$

Step 7.2.2

Multiply $2$ by $8$ .

$\frac{3}{16} u^{8} + C$

Step 8

Replace all occurrences of $u$ with $x^{2} - 3$ .

$\frac{3}{16} {(x^{2} - 3)}^{8} + C$