Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Use logarithm rules to move $\tan (x)$ out of the exponent.

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

Step 1.2

The natural logarithm of $e$ is $1$ .

$\int \sec^{2} (x) (\tan (x) \cdot 1) d x$

Step 1.3

Multiply $\tan (x)$ by $1$ .

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

Step 2

Let $u = \sec (x)$ . Then $d u = \sec (x) \tan (x) d x$ , so $\frac{1}{\sec (x) \tan (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $\sec (x)$ .

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

Step 2.1.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sec (x) \tan (x)$

Step 2.2

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

$\int u d u$

Step 3

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$\frac{1}{2} u^{2} + C$

Step 4

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

$\frac{1}{2} \sec^{2} (x) + C$