1765 words - 7 pages

Improper Integrals

Dr. Philippe B. laval

Kennesaw State University

September 19, 2005

Abstract

Notes on improper integrals.

1 Improper Integrals

1.1 Introduction

In Calculus II, students deﬁned the integral ∫ b a f (x) dx over a ﬁnite interval [a, b].

The function f was assumed to be continuous, or at least bounded, otherwise the integral was not guaranteed to exist. Assuming an antiderivative of f could

be found, ∫ b a f (x) dx always existed, and was a number. In this section, we

investigate what happens when these conditions are not met.

Definition 1 (Improper Integral) An integral is an improper integral if ei- ther the interval of integration is not ﬁnite (improper integral of type 1) or if the function to integrate is not continuous (not bounded) in the interval of integration (improper integral of type 2).

Example 2

∫ ∞

0

e−xdx is an improper integral of type 1 since the upper limit

of integration is inﬁnite.

Example 3

∫ 1

0

dx

x is an improper integral of type 2 because

1

x is not continu-

ous at 0.

Example 4 ∫ ∞

0

dx

x− 1 is an improper integral of types 1 since the upper limit

of integration is inﬁnite. It is also an improper integral of type 2 because 1

x− 1 is not continuous at 1 and 1 is in the interval of integration.

Example 5

∫ 2

−2

dx

x2 − 1 is an improper integral of type 2 because 1

x2 − 1 is not continuous at −1 and 1.

1

Example 6

∫ π 0

tanxdx is an improper integral of type 2 because tanx is not

continuous at π

2 .

We now look how to handle each type of improper integral.

1.2 Improper Integrals of Type 1

These are easy to identify. Simply look at the interval of integration. If either the lower limit of integration, the upper limit of integration or both are not ﬁnite, it will be an improper integral of type 1.

Definition 7 (improper integral of type 1) Improper integrals of type 1 are evaluated as follows:

1. If

∫ t a

f (x) dx exists for all t ≥ a, then we deﬁne

∞∫ a

f (x) dx = lim t→∞

t∫ a

f (x) dx

provided the limit exists as a ﬁnite number. In this case,

∫ ∞

a

f (x) dx is

said to be convergent (or to converge). Otherwise,

∫ ∞

a

f (x) dx is said

to be divergent (or to diverge).

2. If

∫ b t

f (x) dx exists for all t ≤ b, then we deﬁne

b∫ −∞

f (x) dx = lim t→−∞

b∫ t

f (x) dx

provided the limit exists as a ﬁnite number. In this case,

∫ b −∞

f (x) dx

is said to be convergent (or to converge). Otherwise,

∫ b −∞

f (x) dx is

said to be divergent (or to diverge).

3. If both

∫ ∞

a

f (x) dx and

∫ b −∞

f (x) dx converge, then we deﬁne

∞∫ −∞

f (x) dx =

a∫ −∞

f (x) dx+

∞∫ a

f (x) dx

The integrals on the right are evaluated as shown in 1. and 2..

2

1.3 Improper Integrals of...

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