Improper Integral - Improper Integrals Dr Philippe B laval Kennesaw State University Abstract Notes on improper integrals 1 1.1 Improper Integrals

# Improper Integral - Improper Integrals Dr Philippe B laval...

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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 x dx 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 x 2 1 is an improper integral of type 2 because 1 x 2 1 is not continuous at 1 and 1 . 1
Example 6 π 0 tan xdx is an improper integral of type 2 because tan x 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 ).