Prove that: `I_n=int_0^oox^(2n+1)e^-x^2dx=(n !)/2,n in Ndot`

The value of int_0^oox^(2n+1)dote^(-x^2)dx is (n in W)

If [x] denotes the integral part of [x] , show that: int_0^((2n-1)pi) [sinx]dx=(1-n)pi,n in N

int x ^ (2n-1) cos x ^ (n) dx

If I_n=intsqrt((a^2+x^2)^n)dx , Prove that: I_n=(xsqrt((a^2+x^2))^n)/(n+1)+(na^2)/(n+1)int(a^2+x^2)^(n/2-1)dx

If int_(0)^(oo)x^(2n+1)dot e^(-x^(2))dx=360, then the value of n is

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

If I_n=int_0^ooe^(-x)x^(n-1)log_exdx , then prove that I_(n+2)-(2n+1)I_(n+1)+n^2I_n=0

Prove that for ngt1 . int_0^1(cos^-1x)^ndx=n(pi/2)^(n-1)-n(n-1)int_0^1(cos^-1x)^(n-2)dx