Calculus Examples

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

Step 1

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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Step 1.1

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

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Step 1.1.1

Differentiate $\tan (x)$ .

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

Step 1.1.2

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

$\sec^{2} (x)$

Step 1.2

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

$\int u^{2} d u$

Step 2

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

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

Step 3

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

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