Question

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

Answer 1

`I_(n)=int_(0)^(oo) (x^(2))^(n)x e^(-x^(2))dx`
Put `x^(2)=t` or `x dx=dt//2`
`:.I_(n)=1/2 int_(0)^(oo) t^(n)e^(-t)dt`
`=1/2[[-t^(n)e^(-t)]_(0)^(oo)+nint_(0)^(oo) t^(n-1)e^(-t)dt]`
`=1/2[0+n int_(0)^(oo) t^(n-1)e^(-t)dt]`
`=n/2int_(0)^(oo) t^(n-1)e^(-t)dt=nI_(n-1)`
or `I_(n-1)=(n-1)I_(n-2)`
or `I_(n)=n(n-1)(n-2).............1I_(0)`
`n!I_(0)=n! 1/2int_(0)^(oo) e^(-t)dt`
`=n! 1/2 [-e^(-t)]_(0)^(oo) =(n!)/2`