Black-Scholes call via risk-neutral expectation

Derive the Black-Scholes price of a European call by computing a discounted expectation under the risk-neutral measure.

$C=S_0N(d_1)-Ke^{-rT}N(d_2)$. Under the risk-neutral measure $S_T=S_0e^{(r-\sigma^2/2)T+\sigma\sqrt TZ}$, so $C=e^{-rT}\mathbb E[(S_T-K)\mathbf 1_{S_T>K}]$. The strike term is $Ke^{-rT}\mathbb P(S_T>K)=Ke^{-rT}N(d_2)$; for the stock term, completing the square in the Gaussian integral (a shift of $Z$ by $\sigma\sqrt T$) gives $S_0N(d_1)$.