integration of log(cos(x)) is not in JEE syllabus.

Anyhow i am giving u the method to solve it,

cos(x) = (e^ix + e^-ix) / 2

So, I = int log((e^ix + e^-ix) / 2) . dx

Put (e^ix + e^-ix) = y

So, ((i)e^ix + (-i)e^-ix) dx = dy

or, (ie^ix - ie^-ix) dx = dy

or, (i) (e^ix - e^-ix) dx = dy

or, dx = dy / i (4 - y²)

So, I = (1/i) int log( y ) / (4 - y²)^1/2 dy + ((log2) / (i)) int dy / (4 - y²)^1/2

Now to solve first part of integral use polylogarithmic functions.

Finally,

Polylogarithm

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

Note that the similar notation is used for the logarithmic integral.