Calculus Examples

$\tan^{4} (x)$

Step 1

Write $\tan^{4} (x)$ as a function.

$f (x) = \tan^{4} (x)$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int \tan^{4} (x) d x$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $2$ out of $4$ .

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

Step 4.2

Rewrite ${\tan (x)}^{2 (2)}$ as exponentiation.

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

Step 5

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

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

Step 6

Simplify.

$\int 1 - 2 \sec^{2} (x) + \sec^{4} (x) d x$

Step 7

Split the single integral into multiple integrals.

$\int d x + \int - 2 \sec^{2} (x) d x + \int \sec^{4} (x) d x$

Step 8

Apply the constant rule.

$x + C + \int - 2 \sec^{2} (x) d x + \int \sec^{4} (x) d x$

Step 9

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$x + C - 2 \int \sec^{2} (x) d x + \int \sec^{4} (x) d x$

Step 10

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$x + C - 2 (\tan (x) + C) + \int \sec^{4} (x) d x$

Step 11

Simplify the expression.

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Step 11.1

Rewrite $4$ as $2$ plus $2$

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

Step 11.2

Rewrite ${\sec (x)}^{2 + 2}$ as $\sec^{2} (x) \sec^{2} (x)$ .

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

Step 12

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

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

Step 13

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 13.1

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

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Step 13.1.1

Differentiate $\tan (x)$ .

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

Step 13.1.2

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

$\sec^{2} (x)$

Step 13.2

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

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

Step 14

Split the single integral into multiple integrals.

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

Step 15

Apply the constant rule.

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

Step 16

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

$x + C - 2 (\tan (x) + C) + u + C + \frac{1}{3} u^{3} + C$

Step 17

Simplify.

$x - 2 \tan (x) + u + \frac{1}{3} u^{3} + C$

Step 18

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

$x - 2 \tan (x) + \tan (x) + \frac{1}{3} \tan^{3} (x) + C$

Step 19

Add $- 2 \tan (x)$ and $\tan (x)$ .

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

Step 20

The answer is the antiderivative of the function $f (x) = \tan^{4} (x)$ .

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