Page 1
Math 152-copyright Joe Kahlig, 13A
Section 8.9: Improper Integrals.
Improper integrals of Type I
Zt
(a) If
f (x) dx exist for every number t ≥ a then
a
Z∞
Zt
f (x) dx = lim
f (x) dx provided this limit
t→∞
a
a
exists (as a finite number).
Za
(a) If
f (x) dx exist for every number t ≤ a then
Za
Za
f (x) dx = lim
−∞
t
f (x) dx provided this limit
t→−∞
t
exists (as a finite number).
The improper integrals in (a) and (b) are called convergent if the limits exists and divergent if the
limit does not
exist.
∞
a
∞
∞
a
Z
(c) If both
Z
f (x)dx and
a
f (x)dx are convergent, then we define
−∞
Compute these integrals.
Z∞
1.
2
1
dx
(x + 3)1.5
Z
Z
f (x) dx =
−∞
Z
f (x) dx +
a
f (x) dx
−∞
Math 152-copyright Joe Kahlig, 13A
Z∞
2.
0
Z∞
3.
1
cos(x)
dx
1 + sin2 (x)
x+1
dx
(x + 3)(x + 4)
Page 2
Page 3
Math 152-copyright Joe Kahlig, 13A
4. Find the values of p so that these integrals will converge.
Z∞
Z∞
1
dx
xp
1
5
dx
xp
1
Improper integrals of Type II
Zb
(a) If f is continuous on [a, b) and is discontinuous at b, then
Zt
f (x) dx = lim
t→b−
a
f (x) dx provided
a
this limit exists (as a finite number).
Zb
Zb
(a) If If f is continuous on (a, b] and is discontinuous at a, then
f (x) dx provided
f (x) dx = lim
t→a+
a
t
this limit exists (as a finite number).
The improper integrals in (a) and (b) are called convergent if the limits exists and divergent if the
limit does not exist.
Zc
(c) If f has a discontinuity at c where a < c < b, and both
f (x) dx and
a
Zb
then we define
Zc
f (x) dx =
a
Zb
f (x) dx +
a
f (x) dx
c
Zb
f (x) dx are convergent,
c
Math 152-copyright Joe Kahlig, 13A
Compute these integrals.
Z8
5.
1
√
dx
3
x
0
Z1
6.
−3
1
dx
x2
Page 4
Page 5
Math 152-copyright Joe Kahlig, 13A
Comparison Theorem
Suppose that f and g are continuous functions with f (x) ≥ g(x) ≥ 0 for x ≥ a
Z∞
(a) If
Z∞
f (x)dx is convergent, then
a
Z∞
(a) If
g(x)dx is convergent.
a
Z∞
g(x)dx is divergent, then
a
f (x)dx is divergent.
a
Use the comparison theorem to decide if these integrals converge or diverge.
Z∞
7.
√
3
2
Z∞
8.
2
√
3
1
x2 − 1
1
x4
−1
dx
dx
Math 152-copyright Joe Kahlig, 13A
Z∞
9.
4
e−x dx
0
Zπ
10.
0
sin(x)
√
dx
x
Page 6