Asked by diana
Evaluate the improper integral or state that it diverges: integral from -inf to inf (2xe^-x) dx. Please help!
Answers
Answered by
Steve
∫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 -∞
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 -∞