Integral of tan x

Q: How can I check the answer?

Differentiate it: the derivative of -\ln\left(\left|\cos\left(x\right)\right|\right) is \tan\left(x\right). CalcViz checks every antiderivative this way before showing it.

\int \tan\left(x\right)\,dx = -\ln\left(\left|\cos\left(x\right)\right|\right) + C

Write tan x = sin x / cos x and substitute u = cos x: the numerator is −du, which gives −ln|cos x|, equivalently ln|sec x|.

Step by step

Trig rule
∫tan u du = −ln|cos u|.
$\int \tan\left(x\right)\,dx = -\ln\left(\left|\cos\left(x\right)\right|\right)$

f(x) = tan x
F(x) + C

Constant C = 0.0

Every choice of C gives a valid antiderivative, since shifting up or down doesn’t change the slope.

Open in the integral calculator

Questions students ask

What is the integral of tan x?

$\int \tan\left(x\right)\,dx = -\ln\left(\left|\cos\left(x\right)\right|\right) + C$ .

How do you integrate tan x?

Write tan x = sin x / cos x and substitute u = cos x: the numerator is −du, which gives −ln|cos x|, equivalently ln|sec x|.

How can I check the answer?

Differentiate it: the derivative of $-\ln\left(\left|\cos\left(x\right)\right|\right)$ is $\tan\left(x\right)$ . CalcViz checks every antiderivative this way before showing it.