# The Definite Integral

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 the definite integral.