Calculus Examples

$\int \sin (x) \sqrt{\cos (x)} d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

$\int - 1 \sqrt{u} d u$

Step 2

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

$- \int \sqrt{u} d u$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$- \int u^{\frac{1}{2}} d u$

Step 4

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$- (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 5

Rewrite $- (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $- \frac{2}{3} u^{\frac{3}{2}} + C$ .

$- \frac{2}{3} u^{\frac{3}{2}} + C$

Step 6

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

$- \frac{2}{3} {\cos (x)}^{\frac{3}{2}} + C$