**Definite Integration**

In the previous post, our IB Maths Tutors 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

=g(b)-g(a)

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

9.

10. where f(a) is periodic with period ‘a’.

**Walli’s Formula: **

=

Where K = if both m and n are even (m, n N)

= 1 in case the function is odd

Here

**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