Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 4

Cancel the common factor of $\cos (x)$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\int \csc (x) d x$

Step 6

The integral of $\csc (x)$ with respect to $x$ is $\ln (| \csc (x) - \cot (x) |)$ .

$\ln (| \csc (x) - \cot (x) |) + C$