Calculus Examples

$\int 12 \sqrt[3]{\sin (x)} \cos (x) d x$

Step 1

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

$12 \int \sqrt[3]{\sin (x)} \cos (x) d x$

Step 2

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 2.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 2.2

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

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

Step 3

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

$12 \int u^{\frac{1}{3}} d u$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Rewrite $12 (\frac{3}{4} u^{\frac{4}{3}} + C)$ as $12 (\frac{3}{4}) u^{\frac{4}{3}} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Combine $12$ and $\frac{3}{4}$ .

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

Step 5.2.2

Multiply $12$ by $3$ .

$\frac{36}{4} u^{\frac{4}{3}} + C$

Step 5.2.3

Cancel the common factor of $36$ and $4$ .

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Step 5.2.3.1

Factor $4$ out of $36$ .

$\frac{4 \cdot 9}{4} u^{\frac{4}{3}} + C$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 9}{4 (1)} u^{\frac{4}{3}} + C$

Step 5.2.3.2.2

Cancel the common factor.

$\frac{4 \cdot 9}{4 \cdot 1} u^{\frac{4}{3}} + C$

Step 5.2.3.2.3

Rewrite the expression.

$\frac{9}{1} u^{\frac{4}{3}} + C$

Step 5.2.3.2.4

Divide $9$ by $1$ .

$9 u^{\frac{4}{3}} + C$

Step 6

Replace all occurrences of $u$ with $\sin (x)$ .

$9 {\sin (x)}^{\frac{4}{3}} + C$