Integrate 1/(1 + cos^2(x)) dx?

Question

denominator is 1 plus cosine squared (x)

Answer 1

∫1/(1+cos²x)dx

=∫1/(cos²x(sec²x+1))dx

=∫sec²x/(2+tan²x)dx ... as sec²x = 1+tan²x

=∫sec²x/(2(1+tan²x/2)dx

=1/2∫sec²x/(1+((tanx)/√2)²)dx

let u = (tanx)/√2

so du = sec²x/√2

So the integral becomes:

1/2∫1/(1+u²)√2du

=arctanu/√2 + c where c is a constant

=arctan((tanx)/√2)/√2 + c

Answer 2

use the integral of Arctan (u) for this

ArcTan[Tan[x]/Sqrt[2]]/Sqrt[2]