Improper Integrals Definition for Type I and Type II
Type I: Unbounded Domain
Consider a function f that is continuous on an unbounded interval [a, ∞).
∞
Z
t
Z
f (x) dx = lim
Define
f (x) dx provided the limit exists.
t→∞ a
a
Consider a function f that is continuous on an unbounded interval (−∞, b].
b
Z
b
Z
f (x) dx = lim
Define
f (x) dx provided the limit exists.
s→−∞ s
−∞
Consider a function f (x) that is continuous on the entire Real line.
∞
Z
f (x) dx
Define
c
Z
∞
Z
=
f (x) dx +
f (x) dx
−∞
−∞
for some convenient choice of c
c
Z
= lim
s→−∞ s
c
Z
f (x) dx + lim
t→∞ c
t
f (x) dx
provided BOTH converge
Type II: Unbounded Range
Consider f (x) that is continuous on [a, b), but discontinuous at x = b.
b
Z
Define
t
Z
f (x) dx = lim
t→b−
a
f (x) dx provided the limit exists.
a
Consider f (x) that is continuous on (a, b], but discontinuous at x = a.
Z
Define
b
b
Z
f (x) dx = lim
s→a+
a
f (x) dx provided the limit exists.
s
Consider f (x) that is discontinuous at x = c in (a, b).
Z
Define
b
Z
f (x) dx
a
c
=
Z
f (x) dx +
a
f (x) dx
for some convenient choice of c
c
Z
= lim
s→c−
b
s
Z
f (x) dx + lim
a
t→c+
b
f (x) dx
provided BOTH converge
t
• If the Limit exists, then we say the Improper Integral Converges.
• Otherwise, if the Limit Does Not Exist, then we say the Improper Integral Diverges.