Calculus Examples

$\int 15 x^{4} {(3 x^{5} - 9)}^{2} d x$

Step 1

Let $u = 3 x^{5} - 9$ . Then $d u = 15 x^{4} d x$ , so $\frac{1}{15 x^{4}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x^{5} - 9$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x^{5} - 9$ .

$\frac{d}{d x} [3 x^{5} - 9]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x^{5} - 9$ with respect to $x$ is $\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 9]$

Step 1.1.3

Evaluate $\frac{d}{d x} [3 x^{5}]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 9]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$3 (5 x^{4}) + \frac{d}{d x} [- 9]$

Step 1.1.3.3

Multiply $5$ by $3$ .

$15 x^{4} + \frac{d}{d x} [- 9]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$15 x^{4} + 0$

Step 1.1.4.2

Add $15 x^{4}$ and $0$ .

$15 x^{4}$

Step 1.2

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

$\int u^{2} d u$

Step 2

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3} + C$

Step 3

Replace all occurrences of $u$ with $3 x^{5} - 9$ .

$\frac{1}{3} {(3 x^{5} - 9)}^{3} + C$