Calculus Examples

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

Step 1

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

$2 \int \frac{1}{\sqrt[3]{3 x}} d x$

Step 2

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

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Step 2.1.1

Differentiate $3 x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$3 \cdot 1$

Step 2.1.4

Multiply $3$ by $1$ .

$3$

Step 2.2

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

$2 \int \frac{1}{\sqrt[3]{u}} \cdot \frac{1}{3} d u$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{\sqrt[3]{u}}$ by $\frac{1}{3}$ .

$2 \int \frac{1}{\sqrt[3]{u} \cdot 3} d u$

Step 3.2

Move $3$ to the left of $\sqrt[3]{u}$ .

$2 \int \frac{1}{3 \sqrt[3]{u}} d u$

Step 4

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

$2 (\frac{1}{3} \int \frac{1}{\sqrt[3]{u}} d u)$

Step 5

Combine fractions.

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Step 5.1

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

$\frac{2}{3} \int \frac{1}{\sqrt[3]{u}} d u$

Step 5.2

Apply basic rules of exponents.

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Step 5.2.1

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

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

Step 5.2.2

Move $u^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.2.3

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 5.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $2$ by $3$ .

$\frac{6}{3 \cdot 2} u^{\frac{2}{3}} + C$

Step 7.2.3

Multiply $3$ by $2$ .

$\frac{6}{6} u^{\frac{2}{3}} + C$

Step 7.2.4

Cancel the common factor of $6$ .

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Step 7.2.4.1

Cancel the common factor.

$\frac{6}{6} u^{\frac{2}{3}} + C$

Step 7.2.4.2

Rewrite the expression.

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

Step 7.2.5

Multiply $u^{\frac{2}{3}}$ by $1$ .

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

Step 8

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

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