7.2.3 Powerful Notation: The Indefinite Integral

The Fundamental Theorem of Calculus gives added importance to the process of finding antiderivatives. You are probably tired of phrases such as "any antiderivative of `e^(2t)`." The process of antidifferentiation, like that of differentiation, has a notation to replace all those words, and it arises from the Fundamental Theorem, in which antidifferentiation is the most important step in the calculation of a definite integral. We refer to this main step in integration (via the Fundamental Theorem) as indefinite integration, and we give it a notation that suggests "do everything except the calculation with the endpoints":

There is a subtlety in this notation about which we need to be clear: It does not distinguish between some particular antiderivative of `ftext[(]t text[)]` and all the antiderivatives of `ftext[(]t text[)]`. By convention, it usually stands for the latter.

Definition   The indefinite integral of a given function is the family of all functions whose derivatives are the given function.

If `Ftext[(]t text[)]` is any one antiderivative of a given function, then the entire family can be represented by `Ftext[(]t text[)]+C`, where `C` is an arbitrary constant. We illustrate the use of the indefinite integral notation in the following example.

Example 2

Find the indefinite integral of `ftext[(]t text[)]=t^3`.

Solution   ${\displaystyle \int t^{3}dt=\frac{1}{4}{t}^4+C}$

This equation can be read "the antiderivatives of `t^3` are all the functions of the form `t^4text[/]4+C`, where `C` is any constant." But we usually read it "the integral of `t^3` is `t^4text[/]4+C`." The missing adjective here is "indefinite." Indeed, the statement and the notation have nothing directly to do with the definite integral introduced in Section 7.1.

Contrast this with the statement "the integral of `t^3` from `1` to `3` is `20`," which we would write symbolically as

In this case, the missing adjective is "definite," and it is signaled by the endpoints, "from `1` to `3`."

The definite integral of a function over a particular interval is a number — think of area under a curve — whereas the indefinite integral is a whole family of functions — think of parallel curves passing through a slope field. The connecting link between the two is the Fundamental Theorem. Thus, to find ∫ 1 3     t 3   d t , we pick any convenient antiderivative of `t^3`, say, `Gtext[(]t text[)]=t^4text[/]4`, and we calculate

Finally, one more notational convenience will make the preceding calculation even simpler. The expression `Gtext[(]b text[)]-Gtext[(]a text[)]`, which turns up frequently in integral calculations, is often written

which is read, "`Gtext[(]t text[)]` evaluated from `a` to `b`." This notation has the advantage of allowing us to display explicitly the formula for `G`, not just a difference of numerical values. And, as we see in the next example, we don't even have to name our antiderivative.

Example 3

Use the evaluation notation with an antiderivative formula to calculate

Solution   `int_1^3t^4 dt=1/5t^5|_1^3=243/5-1/5=48.4`.

Recall that the notation for the definite integral makes sense as a suggestion of summing infinitely many infinitely small terms of the form `ftext[(]t_ktext[)] Delta t`. The notation for the indefinite integral also makes sense as a description of a procedure by which we can evaluate definite integrals without doing all the hard work of evaluating lengthy sums and finding a limiting value:

`int_a^bftext[(]t text[)] dt =``int  ``ftext[(]t text[)] dt |_a^b`.

The right side of this equation has to be interpreted to mean the same thing as the right side of

that is, find an antiderivative `G` of `f`, and calculate `Gtext[(]b text[)]-Gtext[(]a text[)]`.

The indefinite integral notation is powerful in the sense that it enables us to carry out intricate, multistep calculations without writing down any words, and often without writing many of the intermediate steps. We will see later that the notation is even more powerful than it may now appear, especially when we interpret it in its differential form — using the `dt` — which we have so far ignored.

On the other hand, powerful notation also can be confusing, especially if the concepts it embodies are not yet familiar. The most confusing feature of the indefinite integral and its notation is that it obscures a very important fact:

The definite integral — a limiting value of sums of products of a certain very special type — is a completely different concept from the indefinite integral — a new name for antidifferentiation.

It is only because of a very remarkable fact — the Fundamental Theorem of Calculus — that tradition assigns similar names and notations to these very dissimilar ideas. Now that you have the powerful notations at your disposal, it is your responsibility to associate the appropriate concept with each notation.