Moments of Normal Distribution

The Problem

Let $Z\sim \mathcal{N}(0,1)$ be a standard normal random variable.

1. Compute $\mathbb{E}[Z^n]$ for $n=0,1,2,3,4$.
2. Derive the general formula for $\mathbb{E}[Z^n]$.
3. Let $X=\mu +\sigma Z\sim \mathcal{N}(\mu ,{\sigma }^2)$. Compute $\mathbb{E}[X^3]$ and $\mathbb{E}[X^4]$ (the third and fourth moments of a general normal).