The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 131 Friday, March 12, 2004 12:39 PM

Principles of Calculus 131

INDEFINITE AND IMPROPER INTEGRALS

In the previous section we defined the integral as a real number associ-

ated with a function on an interval (a,b). If we allow the upper limit b to

vary, then the integral defines a function:

x

Fx()= ∫ fu()ud

a

which is called an indefinite integral.

Given a function f, there is an indefinite integral for each starting

point. From the definition of integral, it is immediate to see that any two

indefinite integrals of the same function differ only by a constant. In

fact, given a function f, consider the two indefinite integrals:

x x

F()= fu()du,Fb ()x = ∫ ()du

b

ax ∫ fu

a

If a < b, we can write

x b x

Fa()x = ∫ fu()ud = fu()ud + ∫ fu()ud = constant + Fb ()x

a ∫a b

We can now extend the definition of proper integrals by introducing

improper integrals. Improper integrals are defined as limits of indefinite

integrals either when the integration limits are infinite or when the inte-

grand diverges to infinity at a given point. Consider the improper integral

∞

∫ fx()xd

a

This integral is defined as the limit

∞ x

∫ fx()xd = lim fu()ud

a x → ∞∫ a

if the limit exists. Consider now a function f that goes to infinity as x

approaches the upper integration limit b. We define the improper integral

b

∫ fx()xd

a