2_37 improper integrals.doc

Mathematics Part 2 Mrs. Kristine Sevcenko Improper Integrals Recall the definition of a definite integral: . This definition works only for functions continuous on the segment . If the function has Type II discontinuity ( ) or if a or b (or both) is infinite then summation is impossible. In these occasions we have improper integrals. An improper integral is said to converge if it has a finite value; otherwise it is said to diverge. Type 1 Improper Integrals The function is continuous but one or both integration boundaries are infinite: (a) ; calculated ; (b) ; calculated ; (c) ; his integral is split into two (a) and (b) integrals: where c is any real number; the integral converges if and only if both components converge. If is an antiderivative of and a finite limit exists then the integral converges and . Type 2 Improper Integrals The function has Type II discontinuity at one or both ends of the segment of integration. (a) function is continuous on but ; calculated ; Page 1

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Mathematics Part 2 Mrs. Kristine Sevcenko

Improper Integrals

Recall the definition of a definite integral: . This definition

works only for functions continuous on the segment . If the function has Type II

discontinuity ( ) or if a or b (or both) is infinite then summation is

impossible. In these occasions we have improper integrals.

An improper integral is said to converge if it has a finite value; otherwise it is said to diverge.

Type 1 Improper IntegralsThe function is continuous but one or both integration boundaries are infinite:

(a) ; calculated ;

(b) ; calculated ;

real number; the integral converges if and only if both components converge.

If is an antiderivative of and a finite limit exists then the integral converges

and .

Type 2 Improper IntegralsThe function has Type II discontinuity at one or both ends of the segment of integration.

(a) function is continuous on but ; calculated

;

(b) continuous on but ; calculated ;

where and converges if and only if both components converge.

Page 1

Mathematics Part 2 Mrs. Kristine Sevcenko

If , is an antiderivative of and a finite limit exists then the integral

converges and .

Similarly, if , is an antiderivative of and a finite limit exists then the

integral converges and .

Type 3 Improper IntegralsThey contain both Type II discontinuities and infinite integration boundaries; these integrals can be split into a sum of finite number Type 1 and Type 2 integrals.

Examples. Find whether the given improper integral converges or diverges; if it converges, evaluate it.

1. 2. 3.

1. Type 1; therefore

and the integral diverges.2. Type 1; split into a sum of two integrals:

3. Type 2: ; diverges.

Exercises . Find whether the given improper integral converges or diverges; if it converges, evaluate it.

1. 7. (a is a constant)

2. 8.

3. 9.

4. 10.

5. 11.

6. 12.