Prove that the value of : `int_0^pi(sin(n+1/2)x)/sin(x/2)dx=pi`

The value of int_0^pi (sin(n+1/2)x)/(sin (x/2)) dx is, (a) n in I, n >= 0 pi/2 (b) 0 (c) pi (d) 2pi

Evaluate : int_0^(pi/2)(sin^2x)/(sin x+cos x)dx

Prove that: int_0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx = pi^2

Prove that: int_0^(2pi)(xsin^(2n)x)/(sin^(2n)+cos^(2n)x)dx = pi^2

The value of int_(-pi//2)^(pi//2)(sin^(2)x)/(1+2^(x))dx is

The value of int_(0)^(pi//2) (x+sin x)/(1+cos x)dx , is

Evaluate : int_0^pi(x^2sin2x*sin(pi/2*cosx))/(2x-pi)dx

Prove that: int_0^(pi//2)(sin x)/(sinx-cosx)dx=pi/4

Evaluate: int_0^(pi//4)(sin\ 2x)/(cos^4x+sin^4x)dx

If n is a positive integer, prove that: int_0^(2pi) (cos(n-1)x-cosnx)/(1-cosx)dx=2pi , hence or otherwise, show that int_0^(2pi) (sin((nx)/2)/sin(x/2))^2dx=2npi .