in this entry on Wikipedia stays
$$ \exp\left(\frac{J^2}{2a}\right)\int_{-\infty}^\infty{\exp\left[-\frac{1}{2}a\left(x-\frac{J}{a}\right)^2\right]}dx\quad=\quad\exp\left(\frac{J^2}{2a}\right)\int_{-\infty}^\infty{\exp\left(-\frac{1}{2}ax^2\right)}dx $$ I tried a lot to understand this but still have no idea. Why is this true?
Let us consider a general form $$ I = \int_{-\infty}^\infty f(x+a) \, {\rm d}x. $$ With a change of variables $z=x+a$, we get ${\rm d}z = {\rm d}x$, and $$ I= \int_{-\infty+a}^{\infty+a} f(z) \, {\rm d}z = \int_{-\infty}^\infty f(z) \, {\rm d}z. $$
This change of variables is intuitively a translation of the $x$-axis. But because we are integrating the function over the whole real axis, this translation has no effect.