Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $\cos (x)$ .

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

Step 2.1.2

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

$- \sin (x)$

Step 2.2

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

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

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 4

Simplify the expression.

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Step 4.1

Multiply $- 1$ by $12$ .

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

Step 4.2

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 5

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 6

Simplify.

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Step 6.1

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 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

Multiply $- 12$ by $3$ .

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

Step 6.2.3

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

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Step 6.2.3.1

Factor $4$ out of $- 36$ .

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

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $4$ out of $4$ .

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

Step 6.2.3.2.2

Cancel the common factor.

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

Step 6.2.3.2.3

Rewrite the expression.

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

Step 6.2.3.2.4

Divide $- 9$ by $1$ .

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

Step 7

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

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