April 12, 2002
(1) Evaluate the following improper integrals. Please show all of your work.
(a) 0 + 1/8 = 1/8
We are going to have to find an antiderivative of ln x, so let's start off doing that by integration by parts.
u = ln x dv = dx
du = 1/x dx v = x
Now notice that has an infinite discontinuity at 0. Therefore
+ 0 =
This last limit is an indetrerminate form ∞/∞, so applying L'Hopital's rule gives a final answer of
= [-l] = -1
(2) This limit has indeterminate form 0/0, so we can use L'Hopital's rule to get 3 (0) = 3