Calculus Examples

$\tan (x) + \cot (x)$

Step 1

Write $\tan (x) + \cot (x)$ as a function.

$f (x) = \tan (x) + \cot (x)$

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.

$F (x) = \int \tan (x) + \cot (x) d x$

Step 4

Split the single integral into multiple integrals.

$\int \tan (x) d x + \int \cot (x) d x$

Step 5

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$\ln (| \sec (x) |) + C + \int \cot (x) d x$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 7.2.2

To multiply absolute values, multiply the terms inside each absolute value.

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

Step 8

The answer is the antiderivative of the function $f (x) = \tan (x) + \cot (x)$ .

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