Page 154 - Calculus Demystified

Indeterminate Forms
CHAPTER 5
5.4.2 THE INTEGRAL ON AN INFINITE INTERVAL 141
Let f be a continuous function whose domain contains an interval of the form
[A, +∞). The value of the improper integral

+∞
f(x) dx
A
is defined to be
N

lim f(x) dx.
N→+∞ A
Similarly, we have: Let g be a continuous function whose domain contains an
interval of the form (−∞,B]. The value of the improper integral
B

g(x) dx
−∞
is defined to be
B

lim f(x) dx.
M→−∞ M
EXAMPLE 5.21
Calculate the improper integral

+∞
x −3 dx.
1
SOLUTION
We do this problem by evaluating the limit
N N

lim x −3 dx = lim −(1/2)x −2
N→+∞ 1 N→+∞ 1

= lim −(1/2) N −2 − 1 −2
N→+∞
1
= .
2
We conclude that the integral converges and has value 1/2.
EXAMPLE 5.22
Evaluate the improper integral

−32
x −1/5 dx.
−∞