Calculus Examples

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

Step 1

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

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

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $\tan (x)$ .

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

Step 2.1.2

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

$\sec^{2} (x)$

Step 2.2

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

$\int e^{u} d u$

Step 3

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

$e^{u} + C$

Step 4

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

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