After learning derivatives, it is natural to wonder if there is some inverse operation, a way to get a function given a derivative. This idea exists in the form of the antiderivative, and it is very important for integral calculus. The antiderivative is defined as follows:
Definition: A function F is called an antiderivative of f if F’(x) = f(x)
The definition is rather simple, if the derivative of F(x) is equal to f(x), then F(x) is the antiderivative of f(x). The process of finding an antiderivative however is trickier than finding derivatives.
Example: Suppose that f(x) = x3. Determine F(x), the antiderivative
To solve this problem, we really need to ask, what function’s derivative is x3? It’s clear to see after some trial and error that
These types of antiderivatives are rather easy to calculate. When we don’t have rules such as chain rule and product rule applying, things are straightforward. Let’s look at a few more examples.
Example: Find a function F(x) whose derivative is F’(x) = cos(x)
If we are looking for a function whose derivative is cos(x), we simply need to remember our trigonometric functions.
We know that if f(x) = sin(x), then f’(x) = cos(x). Remember that we need to add a constant to the equation as well. Doing this gives us: F'(x) = sin(x) + C
Example: Find the antiderivative of f(x) = xn, where n > 0.
We saw that when n = 3, the antiderivative was . We can continue to do different examples of powers, and arrive at a general rule:
We will continue to investigate the idea of antiderivatives as we continue through the subject of Integral Calculus. As we continue on, the antiderivatives we compute will become more complex, and in turn, we will see more complex rules for solving them.