Question

Integration of 1+cosx /1+sinx

Answer 1

Multiply numerator and denom by (1-sin x) :

   f(x) dx = [ 1 - sin x + cos x - cos x sin x ] / cos^2 x   dx
         = [ sec^2 x - sec x tan x + sec x - sin x /cos x ] dx
Integrating:
      I  = tan x  - sec x + Ln (sec x + tan x) + Ln cos x  + K
         = - (1 - sin x)/ cos x + Ln (1 + sin x) + K
         =  Tan (π/4 - x/2)  + 2 Ln [ sin (π/4 + x/2) ]  + K1