Prove that, `int_(0)^((pi)/(2))cos^(n)x cos nx dx=(pi)/(2^(n+1))`.

Prove that, int_(0)^(2pi)(xsin^(2n)x)/(sin^(2n)x+cos^(2n)x)dx=pi^(2) .

Prove that, int_(-(pi)/(2))^((pi)/(2))cos^(3)xdx=2int_(0)^((pi)/(2))cos^(3)xdx , hence, find the value of int_(-(pi)/(2))^((pi)/(2))cos^(3)xdx

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

Show that, int_(0)^((pi)/(2))(sin2x dx)/(sin^(4)x+cos^(4)x)=(pi)/(2)

If m and n are integers and m ne n , then show that, int_(0)^(pi) sin mx cos nx dx={{:((2m)/(m^(2)-n^(2)),"when"(m-n)" is odd"),(0,"when"(m-n)"is even"):}

Show that, int_(-(pi)/(2))^((pi)/(2))cos^(5)xdx=2int_(0)^((pi)/(2))cos^(5)xdx and find the value of int_(-(pi)/(2))^((pi)/(2))cos^(5)xdx .

Evaluate : int_0^(pi//2) cos^3xcdot sinx dx

Prove that, int_(0)^(pi)f(sinx)dx=2int_(0)^((pi)/(2))f(sinx)dx .

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

Prove that, int_(-pi)^(pi)(xe^(x^(2)))/(1+x^(2))dx=0