Antiderivative

We now know what derivatives are and how to find them, but now, suppose we know the rate of change (F) of the population (f), and we want to find what the population will be at some point in time. What we have to do is find a function F whose derivative is f. This is known as the antiderivative, and we define it formally as a function F is called an antiderivative of f on  if  for all . 

Suppose we have a function , then  (where c is some constant), from which we can confirm that . 

The following table shows some important functions and their antiderivatives that we will encounter often:

Function	Antiderivative

Let's suppose we have the following function:

We want to find its antiderivative. I know this probably looks like a difficult equation, but by using the preceding table, we can make this very easy for ourselves. Let's see how.

First, we rewrite the function so it becomes the following:

And so, the antiderivative is as follows:

To make things easier, we rewrite this, as follows:

.

And there you have it. 

You may now be wondering whether or not we can find what the value of c is, and if so, how. Let's go through another example, and see how. 

Suppose we have a function that is the second derivative, and we want to find the antiderivative of the antiderivative—that is, the original function. We have the following:

Then, the first antiderivative is as follows:

And so, the second antiderivative is as follows:

Here, we want to find the values of c and d. We can do this simply by plugging in the preceding values and solving for the unknowns, as follows:

We can also do this: 

Thus, our function looks like this: