Prove that :`int_(0)^(pi//2) x . cot x dx =(pi)/(2)log 2`

int_(0)^(pi//2) x^(2) cos x dx

Prove that : int_(0)^(pi//2) (dx)/(1+cot x)=(pi)/(4)

Prove that : int_(0)^(pi) x cos^(2) x dx =(pi^(2))/(4)

Prove that : int_(0)^(pi) sin^(2) x . cos x dx =0

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)(x)/(sin x +cos x)dx= (pi)/(4sqrt(2)) log |(sqrt(2)+1)/(sqrt(2)-1)|

int_(0)^(pi) x log sinx\ dx

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

int_(0)^(pi) x log sinx dx

Prove that : int_(0)^(1)(sin^(-1)x)/(x) dx = (pi)/(2) log 2