Calculus Examples

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

Step 1

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 2

Set up the integral to solve.

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

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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

$- x + C + \tan (x) + C$

Step 7

Simplify.

$- x + \tan (x) + C$

Step 8

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

$F (x) =$ $- x + \tan (x) + C$