Calculus Examples

$\int \sin^{4} (x) \cos^{3} (x) d x$

Step 1

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

$\int \sin^{4} (x) (\cos^{2} (x) \cos (x)) d x$

Step 2

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

$\int \sin^{4} (x) ((1 - \sin^{2} (x)) \cos (x)) d x$

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 $d$ $u$ .

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Step 3.1

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

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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 $d u$ .

$\int u^{4} (1 - u^{2}) d u$

Step 4

Multiply $u^{4} (1 - u^{2})$ .

$\int u^{4} \cdot 1 + u^{4} (- u^{2}) d u$

Step 5

Simplify.

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Step 5.1

Multiply $u^{4}$ by $1$ .

$\int u^{4} + u^{4} (- u^{2}) d u$

Step 5.2

Multiply $u^{4}$ by $u^{2}$ by adding the exponents.

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Step 5.2.1

Move $u^{2}$ .

$\int u^{4} + u^{2} u^{4} \cdot - 1 d u$

Step 5.2.2

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

$\int u^{4} + u^{2 + 4} \cdot - 1 d u$

Step 5.2.3

Add $2$ and $4$ .

$\int u^{4} + u^{6} \cdot - 1 d u$

Step 5.3

Move $- 1$ to the left of $u^{6}$ .

$\int u^{4} - 1 \cdot u^{6} d u$

Step 5.4

Rewrite $- 1 u^{6}$ as $- u^{6}$ .

$\int u^{4} - u^{6} d u$

Step 6

Split the single integral into multiple integrals.

$\int u^{4} d u + \int - u^{6} d u$

Step 7

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5} + C + \int - u^{6} d u$

Step 8

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

$\frac{1}{5} u^{5} + C - \int u^{6} d u$

Step 9

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$\frac{1}{5} u^{5} + C - (\frac{1}{7} u^{7} + C)$

Step 10

Simplify.

$\frac{1}{5} u^{5} - \frac{1}{7} u^{7} + C$

Step 11

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

$\frac{1}{5} \sin^{5} (x) - \frac{1}{7} \sin^{7} (x) + C$