6.1.1 Examples of Antiderivatives

This section is similar in nature to Section 5.6, where we found symbolic derivatives of functions. Here we go in the other direction: We look for antiderivatives, that is, for functions that have a given function as derivative.

Example 1

Find two different functions whose derivative is `ftext[(]t text[)]=t^2.`

Solution   We know by the Power Rule that differentiation of a polynomial decreases the degree by `1.` In particular,

That is close, but not quite what we need. However, the Scalar Multiple Rule says that the derivative of a scalar times a function is the scalar times the derivative. So, to get rid of the `3` that we don't want, we need to multiply by a scalar that cancels it out, i.e., by `1/3`:

So, `gtext[(]t text[)]=1/3t^3` is a function whose derivative is `ftext[(]t text[)]=t^2`.

For a second function, we can add any any non-zero constant, since

Thus `htext[(]t text[)]=1/3t^3+27` also has derivative `ftext[(]t text[)]=t^2` — as does `wtext[(]t text[)]=1/3t^3-1.`

In fact, there is an infinite family of functions with derivative `ftext[(]t text[)]`, namely, any function of the form

Checkpoint 1

In the preceding example and checkpoint we used phrases such as "a function whose derivative is" several times. We need formal terminology for this concept.

Definitions   A function `g text[(]x text[)]` whose derivative is `f text[(]x text[)]` is called an antiderivative of `f text[(]x text[)]` and an antidifferential of `f text[(]x text[)] dx`. The process of finding an antiderivative or antidifferential (i.e., of undoing differentiation) is called antidifferentiation.

We may extend Example 1 to (almost) all power functions, because the Power Rule for derivatives applies to all such functions. Specifically, an antiderivative for `x^r` is `1/(r+1)x^(r+1)` for real numbers `r` except `r=-1.` (Why doesn't this work for `r=-1`?)

Example 2

1. Find an antiderivative for the function `4/sqrt(2x)` for `x>0.`

2. Describe an infinite family of antiderivatives for `4/sqrt(2x)` for `x>0.`

Solution   We know that `d/(dx)sqrt(x)=1/2 1/sqrt(x)=1/(2sqrt(x))`, so, by the Chain Rule,

Now, by the Constant Multiple Rule,

Thus, one antiderivative for `4/sqrt(2x)` is `4sqrt(2x)`, and an infinite family of antiderivatives is

where `C` is any constant.

Checkpoint 2

Example 3

1. Find an antiderivative for the function `sin text[(]3theta text[)]`.

2. Describe an infinite family of antiderivatives for `sin text[(]3theta text[)]`.

Solution   We know that the derivative of the cosine function is minus the sine function. By the Chain Rule, we know that

So, again by the Constant Multiple Rule,

Thus, one antiderivative for `sin text[(]3theta text[)]` is `-1/3cos text[(]3theta text[)],` and an infinite family of antiderivatives is

Checkpoint 3