 Evaluate 
 can be rewritten as Therefore, the given integral becomes

Sometimes, to modify an integral, an appropriate substitution has to be used; the same way we did in the unit on
Indefinite Integration. For example, integrals containing the expression
can be simplified (or modified) using the substitution
.
For evaluating a definite integral too, we can use the appropriate substitution, provided we change the limits of integration accordingly also. This will become clear in subsequent examples.

The substitution can now be used to simplify this integral. However, we must change the limits of integration according to this substitution:
Thus, the modified integral (in terms of the new variable ) is:


In the integral that we are considering, the limits of integration are to i.e.,
In this interval, . Thus,
From property ( ), we can therefore say that:
Using the result of part ( ) for the first and third terms in ( ), we get our desired result:

 For , let . Find the function and show that 

Observe carefully the form of the function : It is in the form of an integral (of another function), with the lower limit being fixed and the upper limit being the variable . As varies, will correspondingly vary.
One approach that you might contemplate to solve this question is evaluate the antiderivative of and then evaluate which will become . However, this will become unnecessarily cumbersome (Try it!). We can, instead, proceed as follows:
Notice that the limits of integration of and are different. If they were the same, we could have added and easily. So we try to make them the same: in , if we let and varies from to will vary from to . This substitution will therefore make the limits of integration of the same as those of :
and can now be easily added:
 
 

We used instead of in . This doesn’t make a difference; is the variable of integration; it can be replaced with any other variable as long as the limits of integration are the same

 
The final expression shows how simplified has become. We let and the limits of integration become to .
Thus,
