Calculus Examples

$\frac{1}{\tan^{2} (x)}$

Step 1

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

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

Step 2

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 3

Set up the integral to solve.

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

Step 4

Simplify.

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Step 4.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{1^{2}}{\tan^{2} (x)} d x$

Step 4.2

Rewrite $\frac{1^{2}}{\tan^{2} (x)}$ as ${(\frac{1}{\tan (x)})}^{2}$ .

$\int {(\frac{1}{\tan (x)})}^{2} d x$

Step 4.3

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

$\int {(\frac{1}{\frac{\sin (x)}{\cos (x)}})}^{2} d x$

Step 4.4

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

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

Step 4.5

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

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

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 $f (x) = \frac{1}{\tan^{2} (x)}$ .

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