How can I evaluate $\lim\limits_{X\to +\infty}\int_{-X}^X x^n \cdot e^{-\frac{x^2}{2}} ~{\rm d}x$?

Question

I have got one exercise which I must solve this integral :

$$\lim_{X\to +\infty}\int_{-X}^Xx^n\cdot e^{-\frac{x^2}{2}} ~{\rm d}x$$

I have got a hint on my book which is :

$$\int_{-X}^Xx^n\cdot e^{-\frac{x^2}{2}}~{\rm d}x=\left[-x^{n-1}e^{-\frac{x^2}{2}}\right]_{-X}^X+(n-1)\int_{-X}^Xx^{n-2}\cdot e^{-\frac{x^2}{2}}~{\rm d}x$$

But I really don't understand How to find a primitive of "$ x\to e^{-\frac{x^2}{2}}$"...??

Answer 1

If $n$ is odd, this integral vanishes for any $X$ due to the integrand being odd, so the limit is also $0$. If $n$ is even, you can use your hint repeatedly to write the integral as several terms of the form $[f(x)]_{-x}^X$ plus a multiple of $\int_{-X}^X e^{-x^2/2}dx$. The terms vanish, again because each $f$ is odd. if you know $\int_\mathbb{R}e^{-x^2/2}dx$, you can finish the calculation of the limit.

Answer 2

Ok thank you

I found $I_{2k}=(2k-1)I_{2k-2}\iff I_{2k}=\dfrac{(2k-1)!}{(k-1)!}\cdot I_0$ with $\displaystyle I_0=\int^{X}_{-X}e^{-\frac{x^2}{2}}\cdot dx$