`int_0^(pi/4) (1+ sin x)/(cos^2 x) dx `

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

If I_(1) = int_(0)^(pi) (x sin x)/(1+cos^2x) dx , I_(2) = int_(0)^(pi) x sin^(4)xdx then, I_(1) : I_(2) is equal to

Evaluate (i) int_0^pi (x sin x)/(1+cos^2 x) dx Evaluate (ii) int_0^pi (4x sin x)/(1+ cos^2 x) dx

int_(0)^( pi)(sin x)/(1+cos^(2)x)dx =

I= int_0^(pi)(x sin x)/(1+cos^(2)x)dx

int_(0)^( pi/2)(sin x)/(1+cos x)dx

(i) int_0^pi (sin^2 x/2- cos^2 x/2) dx (ii) int_0^(pi//2) (sin^2x)/(1+cosx)^2 dx

Evaluate (i) int_0^(pi//4) sin x cos x dx (ii) int_0^(pi//2) (1 + cos x)^(1//2) dx (iii) int_0^(pi//2) (1 + sin x)^(1//2) dx (iv) int_0^(pi//4) (1 -cos 2x)^(1//2)dx

int_(0)^(pi) x sin x. cos^(2) x dx

int_(0)^(pi) x sin x cos^(2)x\ dx