Can u told me about integral and it's properties?
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.
4. Change of variable of integration is immaterial as long as the limits of integration remain the same, i.e.
5. If the limits are interchanged, i.e. the upper limit becomes the lower limit and vice versa, then
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.
Another result that can be derived from this property is
where f(T + x) = f(x), m ∈ I.
where f(x) is periodic with period ‘T’ and n ∈ I.
where f(x) is periodic with period a.
15. If f(x) ≥ 0 on the interval [a, b], then
16. Let f(x) and g(x) be two functions defined and continuous on [a, b], then