Calculus Examples

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

Step 1

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

$\int \cos^{2} (x) \cos (x) \sin^{3} (x) d x$

Step 2

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

$\int (1 - \sin^{2} (x)) \cos (x) \sin^{3} (x) d x$

Step 3

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 3.1

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

Tap for more steps...

Step 3.1.1

Differentiate $\sin (x)$ .

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

Step 3.1.2

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

$\cos (x)$

Step 3.2

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

$\int (1 - u^{2}) u^{3} d u$

Step 4

Expand $(1 - u^{2}) u^{3}$ .

Tap for more steps...

Step 4.1

Apply the distributive property.

$\int 1 u^{3} - u^{2} u^{3} d u$

Step 4.2

Multiply $u^{3}$ by $1$ .

$\int u^{3} - u^{2} u^{3} d u$

Step 4.3

Factor out negative.

$\int u^{3} - (u^{2} u^{3}) d u$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{3} - u^{2 + 3} d u$

Step 4.5

Add $2$ and $3$ .

$\int u^{3} - u^{5} d u$

Step 4.6

Reorder $u^{3}$ and $- u^{5}$ .

$\int - u^{5} + u^{3} d u$

Step 5

Split the single integral into multiple integrals.

$\int - u^{5} d u + \int u^{3} d u$

Step 6

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

$- \int u^{5} d u + \int u^{3} d u$

Step 7

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$- (\frac{1}{6} u^{6} + C) + \int u^{3} d u$

Step 8

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

$- (\frac{1}{6} u^{6} + C) + \frac{1}{4} u^{4} + C$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Combine $\frac{1}{6}$ and $u^{6}$ .

$- (\frac{u^{6}}{6} + C) + \frac{1}{4} u^{4} + C$

Step 9.2

Simplify.

$- \frac{1}{6} u^{6} + \frac{1}{4} u^{4} + C$

Step 10

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

$- \frac{1}{6} \sin^{6} (x) + \frac{1}{4} \sin^{4} (x) + C$