Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$\int \cot (x) d x$

Step 5

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

$\ln (| \sin (x) |) + C$

Step 6

The answer is the antiderivative of the function $f (x) = \frac{1}{\tan (x)}$ .

$F (x) =$ $\ln (| \sin (x) |) + C$