I've got the following integral to resolve:
$$\int_{-\infty}^\pi x\cdot \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}} dx$$
If it were from $-\infty$ to $\infty$, is a first moment of gaussian distribution and i know how to solve. But with this domain of integration I have some doubts. Thank you in advance.
With $z:=\frac{x-\mu}{\sigma}$ we can rewrite your integral as $\int_{-\infty}^a(\mu+\sigma z)\Phi^\prime(z) dz$ with $\Phi$ the $N(0,\,1)$ CDF and $a:=\frac{\pi-\mu}{\sigma}$. We can express this as $$\mu\Phi(a)+\sigma\int_{-\infty}^az\Phi^\prime(z)dz=\mu\Phi(a)-\frac{\sigma}{\sqrt{2\pi}}\exp -\frac{a^2}{2},$$where we have used $z\Phi^\prime=-\Phi^{\prime\prime}$.