homework5 (1) - 31B Homework 5- Due Friday, May 10th in...

31B Homework 5- Due Friday, May 10th in class
Uncollected problems:
Section 8.6 # 5, 6, 11, 20, 21, 27, 29, 37, 40, 43, 47, 59, 64, 67, 77, 81
Section 8.8 # 7, 10, 15, 31, 33, 35, 44, 53
Collected problems:
1. Integrate the following improper integrals:
(a)
Z
1
0
x ln( x ) dx
(b)
Z
∞
-∞
x
3
e
- x
4
dx

Then do a u-sub u = x 4 so du = 4 x 3 dx in both of the integrals to get
Z
∞
-∞
x
3
e
- x
4
dx =
1
4
lim
R →-∞
Z
0
R 4
e
- u
dx + lim
R →∞
Z
R
0
e
- u
dx
= -
1
4
lim
R →-∞
( e
0
- e
- R
4
) + lim
R →∞
( e
- R
4
- e
0
) = 0