Calculus Examples

$\int (\sin^{3} (x)) (\cos (x)) d x$

Step 1

Multiply $\sin^{3} (x)$ by $\cos (x)$ .

$\int \sin^{3} (x) \cos (x) d x$

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 $d$ $u$ .

Tap for more steps...

Step 2.1

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

Tap for more steps...

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 $d u$ .

$\int u^{3} d u$

Step 3

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

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

Step 4

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

$\frac{1}{4} \sin^{4} (x) + C$