Calculus Examples

$\cos^{3} (x)$

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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

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

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $\sin (x)$ .

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

Step 6.1.2

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

$\cos (x)$

Step 6.2

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

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

Apply the constant rule.

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

Step 9

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

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

Step 10

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 11

Simplify.

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

Step 12

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

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

Step 13

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

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