The Indefinite Integral

The Indefinite Integral

How do we undo a derivative? If we were given a function $\displaystyle f(x)$, how could we find a function $\displaystyle F(x)$ whose derivative is $\displaystyle f(x)$? The answer is called the antiderivative of $\displaystyle f(x)$, which we will denote by the associated capital letter $\displaystyle F(x)$.

Another way to think about this question is: "What function do I have to take the derivative of in order to get the answer?" The antiderivative of $\displaystyle f(x) = 2x$ is $\displaystyle x^2$.

But what about $\displaystyle F(x) = x^2 + 1$? This works too! In fact, since when we take the derivative of a constant, we get zero. We could have chosen any constant. As a result, we report our antiderivative in its most general form $\displaystyle x^2 + C$. The constant $\displaystyle C$ is an important part of the antiderivative.

Notationally, we denote the operation of taking the antiderivative (known as integration) as:

This is also called the indefinite integral of the function $\displaystyle f(x)$, or sometimes just the integral of $\displaystyle f(x)$, where $\displaystyle f(x)$ is called the integrand.

That was a pretty simple example, so how do we find antiderivatives of more complicated expressions? In much the same way as we did with derivatives, we can generate a set of rules for finding antiderivatives, derived simply by thinking of our familiar derivative rules in reverse.