# How do you integrate secx?

Sep 2, 2016

The standard trick is to multiply by $\frac{\sec x + \tan x}{\sec x + \tan x}$, then use substitution.

#### Explanation:

$\int \sec x \mathrm{dx} = \int \sec x \frac{\sec x + \tan x}{\sec x + \tan x} \mathrm{dx}$

$= \int \frac{{\sec}^{2} x + \sec x \tan x}{\sec x + \tan x} \mathrm{dx}$

With $u = \sec x + \tan x$, we have $\int \frac{1}{u} \mathrm{du}$, so

$\int \sec x \mathrm{dx} = \ln \left\mid \sec x + \tan x \right\mid + C$