Wats the integral of ln(cosx) ? 

use complex number method for this. however this is not in jee syllabus.

Use, cos(x) = (e^iθ + e^-iθ )/ 2

Now you can integrate it. 

hiiii

where 

The polylogarithm , also known as the Jonquière's function, is the function

           !!!!!

answer>>>>>>(i Li_2(-e^(2 i x)))/2 + (i x^2)/2-x log(1+e^(2 i x))+x log(cos(x))+constant..ohh sorry .

sir has already posted it!!