Welcome to the other half of calculus! This, unfortunately, is the more difficult half, but don’t worry. We’ll
get you through it. In differential calculus, you learned all of the fun things that you can do with the
derivative. Now you’ll learn to do the reverse: how to take an integral. As you might imagine, there’s a
bunch of new fun things that you can do with integrals too.
It’s also time for a new symbol ∫, which stands for integration. An integral actually serves several
different purposes, but the first, and most basic, is that of the antiderivative.
THE ANTIDERIVATIVE
An antiderivative is a derivative in reverse. Therefore, we’re going to reverse some of the rules we
learned with derivatives and apply them to integrals. For example, we know that the derivative of x^2 is
2 x. If we are given the derivative of a function and have to figure out the original function, we use
antidifferentiation. Thus, the antiderivative of 2x is x^2 . (Actually, the answer is slightly more complicated
than that, but we’ll get into that in a few moments.)
Now we need to add some info here to make sure that you get this absolutely correct. First, as far as
notation goes, it is traditional to write the antiderivative of a function using its uppercase letter, so the
antiderivative of f(x) is F(x), the antiderivative of g(x) is G(x), and so on.
The second idea is very important: Each function has more than one antiderivative. In fact, there are an
infinite number of antiderivatives of a function. Let’s go back to our example to help illustrate this.
Remember that the antiderivative of 2x is x^2 ? Well, consider: If you take the derivative of x^2 + 1, you get
2 x. The same is true for x^2 + 2, x^2 − 1, and so on. In fact, if any constant is added to x^2 , the derivative is
still 2x because the derivative of a constant is zero.
Because of this, we write the antiderivative of 2x as x^2 + C, where C stands for any constant.
Finally, whenever you take the integral (or antiderivative) of a function of x, you always add the term dx
(or dy if it’s a function of y, etc.) to the integrand (the thing inside the integral). You’ll learn why later.
Just remember that you must always use the dx symbol, and
teachers love to take points off for forgetting the dx. Don’t ask
why, but they do!Here is the Power Rule for antiderivatives.