(1) Evaluate the following improper integrals. Please show all of your work.

(a)     0 + 1/8 = 1/8


(b)    
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

[-1] +
+ 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