Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\sin (x)$ .

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

Step 1.1.2

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

$\cos (x)$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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