`int_(0)^( pi)x sin^(-1)xdx`

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

int_(0)^( pi/2)x sec^(2)xdx

(i) int_(0)^( pi)x cos^(2)xdx

Prove the following : 0 lt int_(0)^(pi//2)sin^(n+1)xdx lt int_(0)^(pi//2)sin^(2)xdx, n gt 1

Evaluate : (i) int_(0)^(1)sin^(-1)xdx , (ii) int_(1)^(2)(lnx)/(x^(2))dx , (iii) int_(0)^(1)x^(2)sin^(-1)xdx .

The value int_(0)^(pi//2) sin^(2)xdx will be :

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

Evaluate : int_(0)^(pi//2)sin^(8)xdx

(1) int_(0)^( 2pi)cos^(7)xdx

I=int_(0)^( pi)*sec^(2)xdx