To be more specific, can we compute $\int_{-\infty}^{+\infty}x\int e^{-x^2}dxdx $ .
The beginning of this question is:
I've read a paper trying to calculate $\int{P(x_t|x_{t-1})P(x_{t-1}|y)}dx_{t-1}$ while $P(x_t|x_{t-1})$ and $ P(x_{t-1}|y) $ are both Gaussian. Although the writer said this could be compute analytically and exactly, I think the indefinite integral could not be caculate. So I tried to reanalyse the whole derivative process and found that only the mean and covariance of the integral are needed. But I cannot find information about the mean and covariance of the integral of Gaussian on search engine. So I resort for help here. Thanks a lot.