MATH 106B - CALCULUS II
WINTER 2006
QUIZ 7
NAME:
Show ALL your work CAREFULLY.
(a) Determine whether the following improper integral converges or di-
verges by using antiderivative.
Zπ/2
0
cos x
sin xdx
First note that the integral is improper at x=0. By using
substitution with u= sin x,Rcos x
sin xdx =Rdu
u=ln|u|+C=ln|sin x|+C.
It follows that
Zπ/2
0
cos x
sin xdx = lim
b→0Zπ/2
b
cos x
sin xdx
= lim
b→0ln |sin x|
π/2
b
= lim
b→0−ln |sin b|=∞.
Thus the improper integral diverges.
(b) Use comparison to determine whether the following improper integral
converges or diverges.
Z∞
1
dx
x4√2x3+1
Note that for x≥1,x4√2x3+1>x
4so that
0≤Z∞
1
dx
x4√2x3+1
<Z∞
1
dx
x4.
Since R∞
1
dx
x4converges, it follows that our original improper inte-
gral also converges.
Date: March 6, 2006.
1