Changing the limits of integration from $-\infty$ to $\infty$

Question

Is it fair to change from

$$ \int_{-\infty}^{a} \exp \left( -t^2 \right) \mathrm dt $$

replacing $t$ with $-t$ $$ \int_{-a}^{\infty} \exp \left( -t^2 \right) \mathrm dt $$

and thus gaining the advantage of being able to use the complementary error function.

Answer 1

Changing the variable $$t\mapsto -t$$ implies the following change in the integral: $$ \int_{-\infty}^a\exp(-t^2)\,dt=-\int_{\infty}^{-a}\exp(-t^2)\,dt=\int_{-a}^\infty\exp(-t^2)\,dt. $$