Integration of Trig?

Question

Whats the integral of f(x) = cos (ln |sec x|) + 2?

my teacher has me completely stumped...

Answer 1

This one is tricky. Let's use substitution.
Remember that the derivative of ln|sec x| is
tan x. Why? Well it is sec tan x/sec x = tan x.
So let u = ln|sec x| du = tan x dx,
dx = arctan u du.
So now you have
∫ cos(u) arctan u du + 2 ∫ arctan u du.
Here you reach a dead end!
The second integral can be done by parts,
but the first one is not elementary, sorry!
No wonder your teacher stumped you!
The original integral is not elementary.