" 5."quad int_(0)^((pi)/(2))log(cos x)dx

int_(0)^((pi)/(2))log(sin2x)dx

int_(0)^((pi)/(2))cos2x log sin xdx

int_(0)^(pi//2)log (sec x) dx=

int_(0)^(pi) log(1 + cos x)dx .

int_(0)^( pi)log(1+cos x)dx

Prove that, int_(0)^(pi)log(1+cos x)dx=-pi log2 , given int_(0)^((pi)/(2))log((sin x))dx=(pi)/(2)"log"(1)/(2) .

int_(0)^( pi)cos2x*log(sin x)dx

The value of int_(0)^((pi)/(2))log[(3+5cos x)/(3+5sin x)]dx is

int_(0)^((pi)/(2))log(sinx)dx=int_(0)^((pi)/(2))log(cosx)dx=(pi)/(2)log.(1)/(2)