Calculus Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

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

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

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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 7

Simplify.

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