Antiderivatives

What are antiderivatives? Definition of antiderivatives

If f is a function, then we can find another function F such that F' = f. The function F is an antiderivative of f when

F' = f.

In integral calculus, we often want to find the most general function F or a family of functions whose derivative is f.

Example of antiderivatives and how to find antiderivatives in integral calculus

Let f(x) = cos(x).

If we want to find an antiderivative of f(x) or an antiderivative of cos(x), then we think about what function would yield cos(x) when we differentiate.

In the case of cos(x), we know that when we differentiate sin(x), we get cos(x). So, an antiderivative of cos(x) is sin(x). But, if we think about it:

another antiderivative of cos(x) = sin(x) + any constant.

This is because if we differentiate a constant, we get 0. That means the general functions whose derivative is f is:

F(x) = sin(x) + C where C is a constant

The antiderivatives of f(x) = x 2

From the above definition, the antiderivatives of f(x) are any functions F(x) such that F'(x) = f(x). So, we ask ourselves, what can we differentiate to get x ². If you don't know how to do integration yet and you only know how to differentiate, then think of some functions that you know of. Differentiate them in your head to see if you get  x ^2.

Try F(x) = x ³ .

What happens when you differentiate  x ³ ?

The derivative of  x ³  is 3x ² which is not quite x 2 on its own. That means you need to divide the antiderivative by 3. This is illustrated below.

F'(x) = x ³ / 3  + C where C is a constant

To understand the concept of antiderivatives more clearly, read our Free Calculus Help section on Integral and Integration.