Black-Scholes value of a call on a stock power

Under Black-Scholes assumptions (stock price $S$, volatility $\sigma$, rate $r$, no dividends), find the value today of a European option that pays $\max(S_T^{\alpha} - X, 0)$ at time $T$, where $\alpha$ is a positive constant.

$V = e^{-rT}\left[M N(d_1) - X N(d_2)\right]$ with $M = \mathbb{E}[S_T^\alpha] = S_0^\alpha \exp\!\left(\alpha(r - \tfrac12\sigma^2)T + \tfrac12\alpha^2\sigma^2 T\right)$, $d_2 = \frac{\alpha\ln S_0 + \alpha(r-\frac12\sigma^2)T - \ln X}{\alpha\sigma\sqrt T}$ and $d_1 = d_2 + \alpha\sigma\sqrt T$. The key fact is that $S_T^\alpha$ is lognormal with log-volatility $\alpha\sigma$, so the Black formula applies to it directly.