Thomas' Calculus: Early Transcendentals

Overview of Improper Integrals
MAT 104 – Frank Swenton, Summer 2000
Definitions
• A proper integral is a definite integral where the interval is finite and the integrand is defined
and continuous at all points in the interval.
– Proper integrals always converge, that is, always give a finite area
• The trouble spots of a definite integral are the points in the interval of integration that make it
an improper integral, i.e., keep it from being proper. They are of two types:
a. Points where the integrand is undefined or discontinuous
b. ∞ and -∞ are always trouble spots when they appear as limits of integration
• A simple improper integral is an improper integral with only one trouble spot, that trouble spot
being at an endpoint of the interval. Simple improper integrals are defined to be the appropriate
limits of proper integrals, e.g.: Z
1
0
1
x
dx = lim
ε → 0 +
Z
1
ε
1
x
dx
– If the limit exists as a real number, then the simple improper integral is called convergent .
– If the limit doesn’t exist as a real number, the simple improper integral is called divergent .
Dealing with improper integrals
• First step: always locate all trouble spots and split the integral into simple improper integrals, then
deal with the pieces individually.