Calculus Examples

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

Split the single integral into multiple integrals.

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

Apply the constant rule.

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

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

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

Split the single integral into multiple integrals.

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

Apply the constant rule.

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

Simplify.

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Subtract $x$ from $x$ .

$C + 0 + C + \int \sec^{2} (x) d x$

Add $C$ and $0$ .

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

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

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

Simplify.

$\tan (x) + C$