integral -e^2 with variables in limits

Question

I've been looking and maybe not fully understanding gaussian integral but what happens when there are variables in the upper/lower bounds(limits)?

for example:

$$\int_{x^2}^{-x^2} e^{-(x-2t)^2} dt$$

does gaussian integral there still apply here? even if its not $\infty$ to $-\infty$ ?

Answer 1

You can substitute $u=x-2t$ to get $\frac 12\int_a^be^{-u^2}du$ which can be expressed in terms of the error function as $\frac {\sqrt \pi}4 (\operatorname{erf}(b)+\operatorname{erf}(a))$.