Calculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$ .

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

Step 3

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$\frac{1}{3} (\frac{3}{7} x^{\frac{7}{3}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{1}{3} (\frac{3}{7} x^{\frac{7}{3}} + C)$ as $\frac{1}{3} \cdot \frac{3}{7} x^{\frac{7}{3}} + C$ .

$\frac{1}{3} \cdot \frac{3}{7} x^{\frac{7}{3}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{3}$ by $\frac{3}{7}$ .

$\frac{3}{3 \cdot 7} x^{\frac{7}{3}} + C$

Step 4.2.2

Multiply $3$ by $7$ .

$\frac{3}{21} x^{\frac{7}{3}} + C$

Step 4.2.3

Cancel the common factor of $3$ and $21$ .

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Step 4.2.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{21} x^{\frac{7}{3}} + C$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $3$ out of $21$ .

$\frac{3 \cdot 1}{3 \cdot 7} x^{\frac{7}{3}} + C$

Step 4.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 7} x^{\frac{7}{3}} + C$

Step 4.2.3.2.3

Rewrite the expression.

$\frac{1}{7} x^{\frac{7}{3}} + C$