` int cos x^(0)dx`

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

underset equal to int(cos x)/(x)dx=(pi)/(2) then int_(0)^(oo)(cos^(3)x)/(x)dx is

int_(0)^( pi)|cos x|^(3)dx

Prove that int_(0)^(a) f(x) dx = int_(0)^(a) f(a-x) dx . Hence, evaluate int_(0)^(pi//2) (sin x - cos x)/(1+ sin x cos x) dx.

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

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

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

int_(0)^( pi)x*sin x*cos^(4)x*dx

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

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