How do I integrate: $\int_{\mathbb{R}} (S_t - K)^+ \phi(t) dt$ where $\phi$ is a normal density and $S_t$ is a geometric brownian motion?
I know my answer should be $\Phi(d_1)$, where $\Phi$ is the normal CDF and $d_1$ is the $d_1$ appearing in the Black-Scholes price of a European option.