I'm asked to evaluate the following integral:
I know the classic gaussian result, but I cannot seem to use this to help. I'm given a hint to try the substitution: , but I cannot make this work.
Here's my stab at it. Denote the integral by . Because of the symmetry this integral is
Under the change of variable , the integral becomes
.
Now replace the dummy variable with , so
.
Now add giving
Now introduce your hint (sort of)
so
.
Note that gives and gives so with that and noting that
gives .
Thus, (3) becomes
and solving for gives
.
Whew! I think I need to lie down.
In that change of variable, there is indeed a minus sign attached to the dy. But the limits of integration get switched: instead of going from 0 to ∞, the integral goes from ∞ to 0. When you switch the limits back again, that introduces a change of sign, which cancels with the other one.