Finding Antiderivatives

Read this section to see how you can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1-4.

Finding Antiderivatives

In order to use Part 2 of the Fundamental Theorem of Calculus, an antiderivative of the integrand is needed, but sometimes it is not easy to find one. This section collects some of the information we already have about the general properties of antiderivatives and about antiderivatives of particular functions. It shows how to use the information you already have to find antiderivatives of more complicated functions and introduces a "change of variable" technique to make this job easier.


Antiderivatives appear so often that there is a notation to indicate the antiderivative of a function:

Definition: \int \mathrm{f}(x) \mathrm{dx}, read as "the indefinite integral of f" or as "the antiderivatives of f," represents the collection (or family) of functions whose derivatives are f.

If F is an antiderivative of f, then every member of the family \int \mathrm{f}(x) \mathrm{d} \mathrm{x} has the form \mathrm{F}(x)+\mathrm{C} for some constant C. We write \int \mathrm{f}(x) \mathrm{dx}=\mathrm{F}(x)+\mathrm{C}, where C represents an arbitrary constant.

There are no small families in the world of antiderivatives: if f has one antiderivative F, then f has an infinite number of antiderivatives and every one of them has the form \mathrm{F}(x)+\mathrm{C}. This means that there are many ways to write a particular indefinite integral and some of them may look very different. You can check that \mathrm{F}(x)=\sin ^{2}(x), \mathrm{G}(x)=-\cos ^{2}(x), and \mathrm{H}(x)=2 \sin ^{2}(x)+\cos ^{2}(x) all have the same derivative \mathrm{f}(x)=2 \sin (x) \cos (x), so the indefinite integral of 2 \sin (x) \cos (x), \int 2 \sin (x) \cos (x) \mathrm{dx}, can be written in several ways: \sin ^{2}(x)+\mathrm{C} \text {, or }-\cos ^{2}(x)+\mathrm{C}, \text { or } 2 \sin ^{2}(x)+\cos ^{2}(x)+\mathrm{C}.

Practice 1:Verify that \int 2 \tan (x) \cdot \sec ^{2}(x) \mathrm{d} x=\tan ^{2}(x)+\mathrm{C} and \int 2 \tan (x) \cdot \sec ^{2}(x) \mathrm{d} x=\sec ^{2}(x)+\mathrm{C}.

Source: Dale Hoffman,
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