Calculus Examples

$\sin^{3} (x)$

Step 1

Factor out $\sin^{2} (x)$ .

$\int \sin^{2} (x) \sin (x) d x$

Step 2

Using the Pythagorean Identity, rewrite $\sin^{2} (x)$ as $1 - \cos^{2} (x)$ .

$\int (1 - \cos^{2} (x)) \sin (x) d x$

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $\cos (x)$ .

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

Step 3.1.2

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

$- \sin (x)$

Step 3.2

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

$- u + C + \int u^{2} d u$

Step 6

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

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

Step 7

Simplify.

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

Step 8

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

$- \cos (x) + \frac{1}{3} \cos^{3} (x) + C$