Calculus Examples

$\int 20 \sqrt[3]{3 x - 2} d x$

Step 1

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$20 \int \sqrt[3]{3 x - 2} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $3 x - 2$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [- 2]$

Step 2.1.3.3

Multiply $3$ by $1$ .

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

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$3 + 0$

Step 2.1.4.2

Add $3$ and $0$ .

$3$

Step 2.2

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

$20 \int \sqrt[3]{u} \frac{1}{3} d u$

Step 3

Combine $\sqrt[3]{u}$ and $\frac{1}{3}$ .

$20 \int \frac{\sqrt[3]{u}}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$20 (\frac{1}{3} \int \sqrt[3]{u} d u)$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{3}$ and $20$ .

$\frac{20}{3} \int \sqrt[3]{u} d u$

Step 5.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u}$ as $u^{\frac{1}{3}}$ .

$\frac{20}{3} \int u^{\frac{1}{3}} d u$

Step 6

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

$\frac{20}{3} (\frac{3}{4} u^{\frac{4}{3}} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{20}{3} (\frac{3}{4} u^{\frac{4}{3}} + C)$ as $\frac{20}{3} \cdot \frac{3}{4} u^{\frac{4}{3}} + C$ .

$\frac{20}{3} \cdot \frac{3}{4} u^{\frac{4}{3}} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{20}{3}$ by $\frac{3}{4}$ .

$\frac{20 \cdot 3}{3 \cdot 4} u^{\frac{4}{3}} + C$

Step 7.2.2

Multiply $20$ by $3$ .

$\frac{60}{3 \cdot 4} u^{\frac{4}{3}} + C$

Step 7.2.3

Multiply $3$ by $4$ .

$\frac{60}{12} u^{\frac{4}{3}} + C$

Step 7.2.4

Cancel the common factor of $60$ and $12$ .

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Step 7.2.4.1

Factor $12$ out of $60$ .

$\frac{12 \cdot 5}{12} u^{\frac{4}{3}} + C$

Step 7.2.4.2

Cancel the common factors.

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Step 7.2.4.2.1

Factor $12$ out of $12$ .

$\frac{12 \cdot 5}{12 (1)} u^{\frac{4}{3}} + C$

Step 7.2.4.2.2

Cancel the common factor.

$\frac{12 \cdot 5}{12 \cdot 1} u^{\frac{4}{3}} + C$

Step 7.2.4.2.3

Rewrite the expression.

$\frac{5}{1} u^{\frac{4}{3}} + C$

Step 7.2.4.2.4

Divide $5$ by $1$ .

$5 u^{\frac{4}{3}} + C$

Step 8

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

$5 {(3 x - 2)}^{\frac{4}{3}} + C$