Calculus Examples

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

Step 1

Write $\sin^{3} (x) \cos^{2} (x)$ as a function.

$f (x) = \sin^{3} (x) \cos^{2} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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

Step 5

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

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

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $\cos (x)$ .

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

Step 6.1.2

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

$- \sin (x)$

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Rewrite $- 1 u^{2}$ as $- u^{2}$ .

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

Add $2$ and $2$ .

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

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

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

Step 11

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

$- (\frac{1}{3} u^{3} + C) + \int u^{4} d u$

Step 12

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

$- (\frac{1}{3} u^{3} + C) + \frac{1}{5} u^{5} + C$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$- (\frac{u^{3}}{3} + C) + \frac{1}{5} u^{5} + C$

Step 13.2

Simplify.

$- \frac{1}{3} u^{3} + \frac{1}{5} u^{5} + C$

Step 14

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

$- \frac{1}{3} \cos^{3} (x) + \frac{1}{5} \cos^{5} (x) + C$

Step 15

The answer is the antiderivative of the function $f (x) = \sin^{3} (x) \cos^{2} (x)$ .

$F (x) =$ $- \frac{1}{3} \cos^{3} (x) + \frac{1}{5} \cos^{5} (x) + C$