Calculus Examples

$\int \frac{4 \sqrt[3]{x}}{3} d x$

Step 1

Since $\frac{4}{3}$ is constant with respect to $x$ , move $\frac{4}{3}$ out of the integral.

$\frac{4}{3} \int \sqrt[3]{x} d x$

Step 2

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

$\frac{4}{3} \int x^{\frac{1}{3}} d x$

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

$\frac{4}{3} \cdot \frac{3}{4} x^{\frac{4}{3}} + C$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{4 \cdot 3}{3 \cdot 4} x^{\frac{4}{3}} + C$

Step 4.2.2

Multiply $4$ by $3$ .

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

Step 4.2.3

Multiply $3$ by $4$ .

$\frac{12}{12} x^{\frac{4}{3}} + C$

Step 4.2.4

Cancel the common factor of $12$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{12}{12} x^{\frac{4}{3}} + C$

Step 4.2.4.2

Rewrite the expression.

$1 x^{\frac{4}{3}} + C$

Step 4.2.5

Multiply $x^{\frac{4}{3}}$ by $1$ .

$x^{\frac{4}{3}} + C$