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

Prove that int_(0)^(infty)x^n e^(-x) dx=n!., Where n is a positive integer.

Prove that int_(0)^(1)(dx)/(1+x^(n))gt1-(1)/(n)"for n"inN

If int_0^oox^(2n+1)dote^(-x^2)dx=360 , then the value of n is___

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

Show that Gamma(n)=2int_(0)^(infty)e^(-x^(2))x^(2n-1) dx.

Let m,n be two positive real numbers and define f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx and g(m,n)=int_(0)^(1)x^(m-1)(1-m)^(n-1)dx . It is known that f(n) for n gt 0 is finite and g(m, n) = g(n, m) for m, n gt 0. int_(0)^(oo)(x^(m-1))/((1+x)^(m+n))dx=

If I_(n)=int cos^(n)x dx . Prove that I_(n)=(1)/(n)(cos^(n-1)x sinx)+((n-1)/(n))I_(n-2) .

If U_n=int_0^1x^n(2-x)^ndxa n dV_n=int_0^1x^n(1-x)^n dx ,n in N , and if (V_n)/(U_n)=1024 , then the value of n is________