Calculus Examples

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

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

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

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

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

Set up the integral to solve.

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

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

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Differentiate $\cos (x)$ .

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

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

$- \sin (x)$

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

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

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

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

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

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

Rewrite $- (\frac{1}{3} u^{3} + C)$ as $- \frac{1}{3} u^{3} + C$ .

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

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

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

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

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