Integral of sec x

Q: How can I check the answer?

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

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

The classic trick: multiply top and bottom by (sec x + tan x). The numerator becomes the derivative of the denominator, giving ln|sec x + tan x|.

Step by step

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

f(x) = sec 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 sec x?

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

How do you integrate sec x?

The classic trick: multiply top and bottom by (sec x + tan x). The numerator becomes the derivative of the denominator, giving ln|sec x + tan x|.

How can I check the answer?

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