Calculus Examples

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

Step 1

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

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

Step 2

Set up the integral to solve.

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

Step 3

Apply pythagorean identity.

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

Step 4

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

$\int \cot^{2} (x) d x$

Step 5

Using the Pythagorean Identity, rewrite $\cot^{2} (x)$ as $- 1 + \csc^{2} (x)$ .

$\int - 1 + \csc^{2} (x) d x$

Step 6

Split the single integral into multiple integrals.

$\int - 1 d x + \int \csc^{2} (x) d x$

Step 7

Apply the constant rule.

$- x + C + \int \csc^{2} (x) d x$

Step 8

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$- x + C - \cot (x) + C$

Step 9

Simplify.

$- x - \cot (x) + C$

Step 10

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

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