Calculus Examples

$\frac{e^{\tan (x)}}{\cos^{2} (x)}$

Step 1

Write $\frac{e^{\tan (x)}}{\cos^{2} (x)}$ as a function.

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

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.

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

Step 4

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

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

Step 5

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 $d$ $u$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Differentiate $\tan (x)$ .

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

Step 5.1.2

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

$\sec^{2} (x)$

Step 5.2

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

$\int e^{u} d u$

Step 6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{u} + C$

Step 7

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

$e^{\tan (x)} + C$

Step 8

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

$F (x) =$ $e^{\tan (x)} + C$