Math 10b Homework for Sections 5.6 and 5.7
Show all your work on all homework assignments.
I. Section 5.6.
1. Do the following problems on pages 387: # 5, 6, 7, 12, 14, 19, 201, 262, 43.
2. Find the following:
a. Ze
1
x4ln x dx b. Zxsec2x dx c. Zsin(ln x)dx
3. Read Example 6 on page 386. Then do the following:
(a) Do problem #39 on page 388.
(b) Use the reduction formula3you proved in problem #39 to find Z(ln x)3dx.
Show all your work.
II. Section 5.7.
1. Do the following problems on page 394: # 21, 24, 30, 32.
2. Find the following:
(a) Zcos x dx
sin2x−sin x−12 (b) Zx3
x2+ 1 dx (c) Zxtan−1x dx (hint4)
3. A team of biologists stocks a lake with a particular species of fish and estimates that the
carrying capacity of the lake for that species—that is, the maximum number of fish that
the lake can sustain—is 500. This situation gives rise to the integral Z500
P(500 −P)dP .
Use partial fractions to show that
Z500
P(500 −P)dP = ln
P
500 −P
+C.
1Hint: In #20 for §5.6, let u= arctan(1
x).
2Hint: In #26 for §5.6, rewrite the integral as Zt2·te−t2
dt, then start with a substitution.
3Hint: In #3(b) for §5.6, you’ll need to use the reduction formula three times.
4Hint: In #2(c) for §5.7, start with integration by parts.
