Calculus Examples

$\tan^{2} (x) + 1$

Write $\tan^{2} (x) + 1$ as a function.

$f (x) = \tan^{2} (x) + 1$

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

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

Set up the integral to solve.

$F (x) = \int \tan^{2} (x) + 1 d x$

Split the single integral into multiple integrals.

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

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

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

Split the single integral into multiple integrals.

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

Apply the constant rule.

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

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

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

Apply the constant rule.

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

Simplify.

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Simplify.

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Add $- x$ and $x$ .

$C + \tan (x) + C + 0 + C$

Add $C + \tan (x) + C$ and $0$ .

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

Simplify.

$\tan (x) + C$

The answer is the antiderivative of the function $f (x) = \tan^{2} (x) + 1$ .

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