Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\int \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} d x$

Step 1.2

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$\int \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} d x$

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\int \tan (x) \frac{1}{\cos (x)} d x$

Step 1.3.2

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

$\int \tan (x) \sec (x) d x$

Step 2

Since the derivative of $\sec (x)$ is $\tan (x) \sec (x)$ , the integral of $\tan (x) \sec (x)$ is $\sec (x)$ .

$\sec (x) + C$