How to deal with this integration $$\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\exp \left [-\frac{1}{a}\left (\sum_{i=1}^{n}x_{i}+b \right )^2 \right ]\mathrm{d}x_{1}\cdots\mathrm{d}x_{n},\quad a>0\quad .(1)$$
Applying the change of variables formula,let $y_{1}=x_{1},\dots,y_{n-1}=x_{n-1},y_n=\sum_{i=1}^{n}x_{i},$then I get $$(1)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\exp \left [-\frac{1}{a}\left (y_{n}+b \right )^2 \right ]\mathrm{d}y_{1}\cdots\mathrm{d}y_{n}$$ $$=\int_{-\infty}^{\infty}\exp \left [-\frac{1}{a}\left (y_{n}+b \right )^2\right ]\mathrm{d}y_{n}\cdot\coprod_{i=1}^{n-1}\left ( \int_{-\infty}^{\infty}1\mathrm{d}y_{i}\right )=\infty.$$ There must be something wrong! But I can't find it.And how to evaluate this integration?
The first integral (the innermost one):
$$\int_{-\infty}^\infty e^{-\frac1a\left(\sum_{i=1}^n (x_i+b)\right)^2}dx_1=\int_{-\infty}^\infty \sqrt a\, e^{-t^2}dt=\sqrt{a\pi} $$
under the change of variables $\; \frac1{\sqrt a}\sum_{i=1}^n (x_i+b)=t\implies dx_1 =\sqrt a\,dt\;$ . And then all the other integrals are infinity, so...it seems like your result is correct.