When $X\sim N(m,\operatorname{sig}^2)$, i.e. a normal distribution with mean "$m$" and standard deviation "$\operatorname{sig}$" with probability distribution function $($PDF$)~f_{X}(x)$, how can I compute the following integral:
Integral of $\displaystyle\int_{-\infty}^0 \exp(rx)\cdot f_{X}(x)\mathrm dx$
P.S. The answer should be $f\left(\exp\left(mr + \frac{r\cdot\operatorname{sig}^2}2\right)\right)$ . $F\left(-m - \frac{r\cdot(\operatorname{sig}^2}{\operatorname{sig}}\right)$ where $F$ is the standard normal distribution $N(0,1)$.
