Calculus II 
Quiz #7
April 12, 2002
Name____________________    
R.  Hammack 
Score ______


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

(a)    [Graphics:Images/quiz7sol_gr_1.gif][Graphics:Images/quiz7sol_gr_2.gif][Graphics:Images/quiz7sol_gr_3.gif][Graphics:Images/quiz7sol_gr_4.gif][Graphics:Images/quiz7sol_gr_5.gif] 0 + 1/8 = 1/8


(b)    [Graphics:Images/quiz7sol_gr_6.gif]
We are going to have to find an antiderivative of ln x, so let's start off doing that by integration by parts.

[Graphics:Images/quiz7sol_gr_7.gif][Graphics:Images/quiz7sol_gr_8.gif][Graphics:Images/quiz7sol_gr_9.gif][Graphics:Images/quiz7sol_gr_10.gif][Graphics:Images/quiz7sol_gr_11.gif]
  u = ln x         dv = dx
du = 1/x dx       v = x

Now notice that [Graphics:Images/quiz7sol_gr_12.gif]has an infinite discontinuity at 0. Therefore
[Graphics:Images/quiz7sol_gr_13.gif][Graphics:Images/quiz7sol_gr_14.gif][Graphics:Images/quiz7sol_gr_15.gif][Graphics:Images/quiz7sol_gr_16.gif][Graphics:Images/quiz7sol_gr_17.gif][Graphics:Images/quiz7sol_gr_18.gif][Graphics:Images/quiz7sol_gr_19.gif]
[Graphics:Images/quiz7sol_gr_20.gif][Graphics:Images/quiz7sol_gr_21.gif][Graphics:Images/quiz7sol_gr_22.gif][-1][Graphics:Images/quiz7sol_gr_23.gif][Graphics:Images/quiz7sol_gr_24.gif][Graphics:Images/quiz7sol_gr_25.gif] +[Graphics:Images/quiz7sol_gr_26.gif][Graphics:Images/quiz7sol_gr_27.gif]
[Graphics:Images/quiz7sol_gr_28.gif][Graphics:Images/quiz7sol_gr_29.gif][Graphics:Images/quiz7sol_gr_30.gif] + 0 = [Graphics:Images/quiz7sol_gr_31.gif][Graphics:Images/quiz7sol_gr_32.gif]
This last limit is an indetrerminate form ∞/∞, so applying L'Hopital's rule gives a final answer of
[Graphics:Images/quiz7sol_gr_33.gif][Graphics:Images/quiz7sol_gr_34.gif]= [Graphics:Images/quiz7sol_gr_35.gif][-l] = -1

(2)  
This limit has indeterminate form 0/0, so we can use L'Hopital's rule to get [Graphics:Images/quiz7sol_gr_36.gif][Graphics:Images/quiz7sol_gr_37.gif][Graphics:Images/quiz7sol_gr_38.gif][Graphics:Images/quiz7sol_gr_39.gif]3 [Graphics:Images/quiz7sol_gr_40.gif](0) = 3