Exact expected maxima

Let $Z_i$ be iid $N(0,1)$ and $M_N=\max_{i\le N}Z_i$. (a) Show that $\mathbb E[M_N]=\int_{-\infty}^{\infty}x\,N\phi(x)\Phi(x)^{N-1}dx$ and, integrating by parts with $x\phi(x)=-\phi'(x)$, that $\mathbb E[M_N]=N(N-1)\int_{-\infty}^{\infty}\phi(x)^2\Phi(x)^{N-2}dx$ for $N\ge2$. (b) Prove $\mathbb E[M_2]=1/\sqrt\pi$ and $\mathbb E[M_3]=3/(2\sqrt\pi)$. (c) Compare with the Gumbel approximation (4.17) for $N=2,3$ and explain the error.