Calculus Examples

$\int 12 x \sqrt[3]{x} d x$

Step 1

Simplify.

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Step 1.1

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

$\int 12 x \cdot x^{\frac{1}{3}} d x$

Step 1.2

Multiply $x$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 1.2.1

Move $x^{\frac{1}{3}}$ .

$\int 12 (x^{\frac{1}{3}} x) d x$

Step 1.2.2

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

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Step 1.2.2.1

Raise $x$ to the power of $1$ .

$\int 12 (x^{\frac{1}{3}} x^{1}) d x$

Step 1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 12 x^{\frac{1}{3} + 1} d x$

Step 1.2.3

Write $1$ as a fraction with a common denominator.

$\int 12 x^{\frac{1}{3} + \frac{3}{3}} d x$

Step 1.2.4

Combine the numerators over the common denominator.

$\int 12 x^{\frac{1 + 3}{3}} d x$

Step 1.2.5

Add $1$ and $3$ .

$\int 12 x^{\frac{4}{3}} d x$

Step 2

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{\frac{4}{3}} d x$

Step 3

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

$12 (\frac{3}{7} x^{\frac{7}{3}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $12 (\frac{3}{7} x^{\frac{7}{3}} + C)$ as $12 (\frac{3}{7}) x^{\frac{7}{3}} + C$ .

$12 (\frac{3}{7}) x^{\frac{7}{3}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $12$ and $\frac{3}{7}$ .

$\frac{12 \cdot 3}{7} x^{\frac{7}{3}} + C$

Step 4.2.2

Multiply $12$ by $3$ .

$\frac{36}{7} x^{\frac{7}{3}} + C$