[tex]\displaystyle \int\sec x\:dx = \ln |\sec x + \tan x| + C[/tex]
Step-by-step explanation:
[tex]\displaystyle \int\sec x\:dx=\int\sec x\left(\frac{\sec x+ \tan x}{\sec x + \tan x}\right)dx[/tex]
[tex]\displaystyle = \int \left(\dfrac{\sec x\tan x + \sec^2x}{\sec x + \tan x} \right)dx[/tex]
Let [tex]u = \sec x + \tan x[/tex]
[tex]\:\:\:\:\:\:du = (\sec x\tan x + \sec^2x)dx[/tex]
where
[tex]d(\sec x) = \sec x\tan x\:dx[/tex]
[tex]d(\tan x) = \sec^2x\:dx[/tex]
[tex]\displaystyle \Rightarrow \int \left(\frac{\sec x\tan x + \sec^2x}{\sec x + \tan x}\right)\:dx = \int \dfrac{du}{u}[/tex]
[tex]= \ln |u| + C = \ln |\sec x + \tan x| + C[/tex]
