Prove that `int_0^(pi/2) sin2xlogtanxdx=0`

int_0^(pi/2) sin^2xcos^2xdx

Prove that: int_0^(pi//2)log|tanx+cotx|dx=pi(log)_e2

Prove that for any positive integer k ,(sin2k x)/(sinx)=2[cosx+cos3x++cos(2k-1)x]dot Hence, prove that int_0^(pi/2)sin2x kcotxdx=pi/2dot

Prove that int_0^(pi/2)sin^3xdx=2/3

int_0^(2pi) sin^2x dx

Prove that : int_0^(pi/2)sqrt(1-sin2x)dx=2(sqrt(2)-1)

Prove that: int_0^(pi//2)logsinx\ dx=\ int_0^(pi//2)logcosx\ dx=-pi/2log2

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

Evaluate int_0^(pi/2) sin^2x dx

Prove that int_(0)^(pi//2)log (sinx)dx=int_(0)^(pi//2) log (cosx)dx=-(pi)/(2) log 2 .