`int_(0)^(pi).log sin^(2)x dx =`

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

The value of the integral int_(0)^(pi)x log sin x dx is

Which of the following are false : Statement-I : ( int_(0)^(pi//2) (sqrt(cos x))/(sqrt(cos x + sqrt(sin x)))= pi/2 Statement-II : int_(0)^(pi//2) log(tan x) dx=1 Statement-III: int_(0)^(pi//2) log sin x dx = - pi log 2

If I_(1)=int_(0)^(pi//2)log (sin x)dx and I_(2)=int_(0)^(pi//2)log (sin 2x)dx , then

int_(0)^(1)log sin((pi)/(2)x)dx equals

Show that int_(0)^(pi//2) log (sin x) dx = - pi/2 log 2

Statement-1: int_(0)^(pi//2) x cot x dx=(pi)/(2)log2 Statement-2: int_(0)^(pi//2) log sin x dx=-(pi)/(2)log2

Prove that: int_(0)^(pi//2) log (sin x) dx =int_(0)^(pi//2) log (cos x) dx =(-pi)/(2) log 2