Calculus Examples

$\int \tan^{3} (x) \sec (x) d x$

Step 1

Raise $\sec (x)$ to the power of $1$ .

$\int \tan^{3} (x) \sec^{1} (x) d x$

Step 2

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

$\int \tan^{2} (x) \tan (x) \sec^{1} (x) d x$

Step 3

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$\int (- 1 + \sec^{2} (x)) \tan (x) \sec^{1} (x) d x$

Step 4

Simplify.

$\int (- 1 + \sec^{2} (x)) \tan (x) \sec (x) d x$

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $\sec (x)$ .

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

Step 5.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 5.2

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Apply the constant rule.

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

Step 8

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 9

Simplify.

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

Step 10

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

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