Important Properties of Definite Integrals Commerce Notes | EduRev

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Definite integration is an important component of integral calculus which generally fetches a good number of questions in various competitive exams. Both indefinite and definite integration are interrelated and indefinite integration lays the groundwork for definite integral. Students are advised to learn all the important formulae as they aid in answering the questions easily and accurately. We shall discuss here some of the important properties of definite integrals and then throw some light on their applications as well: 

Important Properties of Definite Integrals

Some of the basic yet important properties which prove fruitful while attempting questions on definite integral are listed below. These results can be proved easily, but the derivations are not very important. Students must remember all the results as the questions can’t be solved unless you know these results.

1. Important Properties of Definite Integrals Commerce Notes | EduRev

2. Important Properties of Definite Integrals Commerce Notes | EduRev

3. Important Properties of Definite Integrals Commerce Notes | EduRev

4. Change of variable of integration is immaterial as long as the limits of integration remain the same, i.e.

Important Properties of Definite Integrals Commerce Notes | EduRev

5. If the limits are interchanged, i.e. the upper limit becomes the lower limit and vice versa, then

 Important Properties of Definite Integrals Commerce Notes | EduRev

6. If f is a piecewise continuous function, then the integral is broken at the points of discontinuity or at the points where the definition of f changes, i.e.

Important Properties of Definite Integrals Commerce Notes | EduRev

7. Important Properties of Definite Integrals Commerce Notes | EduRev

                         Important Properties of Definite Integrals Commerce Notes | EduRev                    

                         Important Properties of Definite Integrals Commerce Notes | EduRev

8. Important Properties of Definite Integrals Commerce Notes | EduRev

Another result that can be derived from this property is

Important Properties of Definite Integrals Commerce Notes | EduRev

9. Important Properties of Definite Integrals Commerce Notes | EduRev

                          Important Properties of Definite Integrals Commerce Notes | EduRev

                          Important Properties of Definite Integrals Commerce Notes | EduRev

10. Important Properties of Definite Integrals Commerce Notes | EduRev

where f(T + x) = f(x), m ∈ I.

11. Important Properties of Definite Integrals Commerce Notes | EduRev

12. Important Properties of Definite Integrals Commerce Notes | EduRev

where f(x) is periodic with period ‘T’ and n ∈ I.

13. Important Properties of Definite Integrals Commerce Notes | EduRev

where f(x) is periodic with period a.

14. Important Properties of Definite Integrals Commerce Notes | EduRev

15. If f(x) ≥ 0 on the interval [a, b], then 

Important Properties of Definite Integrals Commerce Notes | EduRev

16. Let f(x) and g(x) be two functions defined and continuous on [a, b], then 

Important Properties of Definite Integrals Commerce Notes | EduRev

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