Calculus Examples

$\int 3 x^{2} {(x^{3} - 4)}^{8} d x$

Step 1

Let $u = x^{3} - 4$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3 x^{2}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} - 4$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} - 4$ .

$\frac{d}{d x} [x^{3} - 4]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 4]$

Step 1.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

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

$\int u^{8} d u$

Step 2

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

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

Step 3

Replace all occurrences of $u$ with $x^{3} - 4$ .

$\frac{1}{9} {(x^{3} - 4)}^{9} + C$