Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Apply the distributive property.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$- (\ln (| \sec (x) |) + C) + \int \sec^{2} (x) \tan (x) d x$

Step 7

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 7.1

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

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Step 7.1.1

Differentiate $\sec (x)$ .

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

Step 7.1.2

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

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

Step 7.2

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

$- (\ln (| \sec (x) |) + C) + \int u d u$

Step 8

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

$- (\ln (| \sec (x) |) + C) + \frac{1}{2} u^{2} + C$

Step 9

Simplify.

$- \ln (| \sec (x) |) + \frac{1}{2} u^{2} + C$

Step 10

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

$- \ln (| \sec (x) |) + \frac{1}{2} \sec^{2} (x) + C$