Calculus Examples

$\frac{\cos (x)}{1 - \cos^{2} (x)}$

Write $\frac{\cos (x)}{1 - \cos^{2} (x)}$ as a function.

$f (x) = \frac{\cos (x)}{1 - \cos^{2} (x)}$

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Set up the integral to solve.

$F (x) = \int \frac{\cos (x)}{1 - \cos^{2} (x)} d x$

Simplify.

Tap for more steps...

Apply pythagorean identity.

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

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

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

Separate fractions.

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

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

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

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

$\int \csc (x) \cot (x) d x$

Since the derivative of $- \csc (x)$ is $\csc (x) \cot (x)$ , the integral of $\csc (x) \cot (x)$ is $- \csc (x)$ .

$- \csc (x) + C$

The answer is the antiderivative of the function $f (x) = \frac{\cos (x)}{1 - \cos^{2} (x)}$ .

$F (x) =$ $- \csc (x) + C$