Improper Integrals
Section 5.10
The definition of the definite integral requires the interval over which we are integrating to be finite. The
Evaluation Theorem further requires the integrand to be a continuous function over the interval we are
integrating. Integrals with infinite intervals of integration and integrals with integrands that have an infinite
discontinuity (i.e vertical asymptote) at some point on the interval of integration are called improper
integrals.
INTEGRALS WITH INFINITE INTERVALS OF INTEGRATION (TYPE 1)
If fis continuous for xa, then a
fxdx lim
t→a
tfxdx (provided the limit exists)
If fis continuous for xb, then −
bfxdx lim
t→− t
bfxdx (provided the limit exists)
If one of the above limits exist, we say the corresponding improper integral CONVERGES. If the limit
does not exist (or is infinite), we say the improper integral DIVERGES.
If fis continuous for all real numbers, then −
fxdx −
afxdx a
fxdx provided both the
integrals on the left converge.
Examples:
1.1
1
x2dx
2.1
1
xdx
3.1
1
xpdx
4.−
0exdx
5.1
1−xe−xdx
6.−
ex
1e2xdx
-3 -2 -1 1 2 3
0.2
0.4
0.6
x
y
Graph of yex
1e2x
