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

Answer

6

Explanation

`I=int_(0)^(oo) (x^(2))^(n).xe^(-x^(2)dx`
Put `x^(2)=t` and `x dx=dt//2`
`=1/2[-t^(n)e^(-t)]_(0)^(oo) +nint_(0)^(oo) t^(n-1)e^(-t)dt]`
`=1/2[0+nint_(0)^(oo) t^(n-1)e^(-t)dt]`
`=(n!)/2=360`
`impliesn=6`