Calculus Examples

$\sec^{2} (x) \tan^{2} (x)$

Write $\sec^{2} (x) \tan^{2} (x)$ as a function.

$f (x) = \sec^{2} (x) \tan^{2} (x)$

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.

$F (x) = \int \sec^{2} (x) \tan^{2} (x) d x$

Let $u = \tan (x)$ . Then $d u = \sec^{2} (x) d x$ , so $\frac{1}{\sec^{2} (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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

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Differentiate $\tan (x)$ .

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

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

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

$\int u^{2} d u$

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

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

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

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

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

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