Properties of Definite Integrals





we are going to see six properties in this page properties of definite integrals.Using these properties we can easily evaluate integrals. Here you can find example problems to understand this topic more clearly.

Property 1:

Which means integration is independent of change of variables provided the limits of integration remain the same.


Property 2:

If the limits of definite integral are interchanged then the values of integrations changes the sign only not value.

Property 3:

Property 4:

Property 5:

Property 6:

Property 7:

Property 8:

This property is having two subdivisions. If the given function is even then we can apply the following property.

If the given function is odd then we have to apply the following property

Example 1:

Evaluate the following

Solution:

Let the given function be f(x). In this problem we have to use the property 8. Because we have same number for both upper and lower limits. So we have to check whether it is odd or even function. For that we have to plug -x instead of x in the given function

 f(x) = log [ (3-x)/(3+x) ]

This looks like the formula log(m/n) .So we can expand this by log m - log n

  f (x) = log [ (3-x) - (3+x) ]

  f(-x) = log [ (3- (-x) - (3 + (-x)) ]

  f(-x) = log [ (3+x) - (3 - x) ]

  f(-x) = log [- (- (3+x) + (3 - x)) ]

  f(-x) = log [- (3 - x) - (3+x) ]

  f(-x) = - log [ (3 - x) - (3+x) ]

  f(-x) = - log [ (3 - x)/ (3+x) ]

  f(-x) = - f(x)

Therefore the given function is odd.

So the answer for the given problems is 0.

Example 2:

Evaluate the following

Solution:

Let the given function be f(x). In this problem we have to use the property 8. Because we have same number for both upper and lower limits. So we have to check whether it is odd or even function. For that we have to plug -x instead of x in the given function

 f(x) = x sin x

  x = -x

f(-x) = (-x) sin (-x)

f(x) = x sin x

Therefore we can say the given function is even function.





Properties of Definite Integrals to Integration