Evaluate the improper integral or state that it diverges: integral from -inf to inf (2xe^-x) dx. Please help!

Answer 1

∫2x e^-x dx = -2 e^-x (x+1) = -2(x+1)/e^x

Now, e^x grows so much faster than x that the limit is zero as x -> ∞.

That can be justified by using lHospital's Rule, to show that the limit is the same as

-2/e^x -> 0

However, as x -> -∞ e^x grows very fast, so the limit is -∞