Consider an expression such that
Integrating both sides with respect to , we have
Putting the value in equation (i), we have
Similarly, putting the value in equation (i), we have
Subtracting from , we have
We can write the above expression as
Since is the anti-derivative , so , where we have ignored as it is cancelled out in (ii). Making this substitution in (ii), we have
We conclude that if is the anti-derivative of , i.e.
Then it can be written as
Thus, we have
The integral is known as a definite integral.