connection between moment of normal r.v to gamma function

Question

Problem: $X \sim \mathcal N(0,1)$ and $p>1 $. Prove that $E[X^p]=\sqrt{2^p} \times \Gamma\left(\frac{p}{2}+\frac{1}{2}\right)/\Gamma({1}/{2})$.

I tried calculating the pdf of $X^p$ and then calculating the expectation using $\int xf(x) dx$ and show that it is equal to the right part but haven't managed to get it.

Is there a diffrent approach I should try?

Answer 1

Yes, use the law of the unconscious statistician and calculate $\int_{\mathbb{R}}\phi(x)x^p dx$, where $\phi(x)$ is the probability density of the standard normal.