In the previous post, we discussed indefinite integration. Now we shall discuss definite integration
► Definite Integration- We already know that this c here is an integral constant. we are not sure about its value. This c is the reason we call this process indefinite integration. But suppose we do our integration between certain limits like:-
here a lower limit while b higher limit
You can clearly see that this function is independent of ‘c’. Means we can be sure about its value so this type of integration is called Definite Integration.
►Definite Integration of a function f(x) is possible in [a,b] if f(x) is continuous in the given interval
►If f(x), the integrand, is not continuous for a given value of x then it doesn’t mean that g(x), the integral, is also discontinuous for that value of x.
► Definite integration of a function between given limits like Algebraic sum of areas bounded by the given curve f(x) and given lines x=a and x=b. That’s why the answer for definite integration problems is a single number.
► If that shows a few things:-
(i) The lines between which area is bounded are co-incident(a=b)
(ii) Area covered above the x-axis=Area covered below the x-axis that means positive part of area and negative part of area is equal
(iii) there must be at least one solution/root to f(x) between x=a and x=b(this is something we study in ROLE’S THEOREM in detail)
► If given function f(x) is not continuous at x=c then we should write
► If given function f(x) > or <0 in any given interval (a,b) then >0 or <0 in given interval (a,b)
► If given function f(x) g(x) in the given interval (a,b) then
in the given interval
► If we integrate the given function f(x) in the given interval (a,b) then
Some More Properties of Definite Integration:-
1. We can interchange the limits on any definite integral, all that we need to do is tack a minus sign onto the integral when we do.
2. . If the upper and lower limits are the equal then integration of function will be zero
3. , where c is any constant/any real number
4. that means definite integration is a distributive process
5. here c is a number lying somewhere between a and b
6. If we don’t change the integrand and the limits, then change in the variable will not affect the answer
7.(a) If f(x) is an odd function i.e. f(x) = – f(-x) then
(b) If f(x) is an even function i.e. f(x) = f(-x) then
8. in particular
10. where f(a) is periodic with period ‘a’.
Where K = if both m and n are even (m, n N)
= 1 in case the function is odd
Leibnitz’s Rule- If f(x) is a continuous function and u(x) & v(x) are differentiable in the interval [a,b] then,
This rule is used when at least one of the limits is a function.
Here is a very detailed Pdf for definite integration download it solves the questions