Evaluate `underset(0)overset(pi/2)int (sin x)/(1+cos^(2)x)dx`

underset(0)overset(pi//4)int log ((sin x+cos x)/(cos x))dx

Prove that int_(0)^(a) f(x)dx=int_(0)^(a) f(a-x)dx and hence evaluate underset(0)overset(pi/2)int (cos^(5)x)/(sin^(5)+x+cos^(5)x)dx .

Prove that underset(-a)overset(a)int f(x)dx={{:(,2underset(0)overset(a)int f(x)dx,"if f(x) is an even function"),(,0,"if f(x) is an odd function"):} and hence evaluate underset(-pi//2)overset(pi//2)int (x^(3)+x cos x)dx.

The value of the intergral underset(0)overset(pi//2)(int) (sin^(100) x-cos ^(100) x) dx is :

Evaluate : underset(2)overset(3)int (xdx)/(x^(2)+1)

int(sin(2x)dx)/(1+cos^(2)x)=

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

Evaluate underset(0)overset(2)int (x^(2)+1)dx as a limit of a sum.

The value of the integral intunderset(0)overset(pi)(x sin^(2n)x)/(sin^(2x)x+cos^(2n)x) dx is