Math 2924 Improper Integrals - Arkansas Tech University...

Arkansas Tech University
MATH 2924: Calculus II
Dr. Marcel B. Finan
Solutions to Assignment 6.6
Exercise 1
(a) This is an improper integral of type 2 where the integrand is discontin-
uous at x = 1 .
(b) This is an improper integral of type 1 where the interval of integration
is unbounded.
(c) This is an improper integral of type 1 where the interval of integration
is unbounded.
(d) This is an improper integral of type 2 where the integrand is discontin-
uous at x = 0
Exercise 5
First, letting u = x - 2 , we find
Z
dx
( x - 2)
3
2
=
Z
u
-
3
2
du = -
2
√
u
= -
2
√
x - 2
+ C.
Thus,
Z
∞
3
dx
( x - 2)
3
2
= lim
b →∞
2 -
2
√
b - 2
= 2
so that the improper integral is convergent
Exercise 7
First, letting u = 3 - 4 x, we find
Z
dx
3 - 4 x
= -
1
4
Z
du
u
= -
1
4
ln | u | = -
1
4
ln | 3 - 4 x | + C.
Thus,
Z
0
-∞
dx
3 - 4 x
= lim
a →-∞
-
ln 3
4
+
1
4
ln | 3 - 4 a | = ∞
so that the improper integral is divvergent
1

Exercise 13
First, letting u = x
2
, we find
Z
xe
- x
2
dx =
1
2
Z
e
- u
du = -
1
2
e
- u
= -
1
2
e
- x
2
+ C.