A tale of three integrals

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At this point we have three “different” integrals.

At this point we have three different “integrals.” Let’s see if we can sort out the differences.

1 Indefinite integrals

An indefinite integral, also called an antiderivative computes classes of functions:

$\int f(x) dx = \text {``a class of functions whose derivative is $f$''}$

Here there are no limits of integration, and your answer will have a “ $+C$ ” at the end. Pay attention to the notation:

Where $F'(x) = f(x)$ .

Indefinite integrals limits of integration, and they compute .

Two students, say Devyn and Riley, are working with the following indefinite integral:

$\int \frac {2}{x\ln (x^2)} dx$

Devyn computes the integral as

$\int \frac {2}{x\ln (x^2)} dx = \ln |\ln |x^2|| + C$

and Riley computes the integral as

$\int \frac {2}{x\ln (x^2)} dx = \ln |\ln |x|| + C.$

Which student is correct?

Both students are correct! The seeming discrepancy arises from the fact that the “+C” in each case is different!

2 Accumulation functions

An accumulation function, also called an area function computes accumulated area:

$\int _a^x f(t) dt = \text {``a function $F$ whose derivative is $f$''}$

This is a function of $x$ whose derivative is $f$ , with the additional property that $F(a)=0$ . Pay attention to the notation:

Where $F'(x) = f(x)$ .

Accumulation functions limits of integration, and they compute .

True or false: There exists a function $f$ such that

$\int _0^x f(t) dt = e^x$

Let

$F(x) = \int _0^x f(t) dt,$

this is an accumulation function and $F(0) = 0$ , since no area is accumulated yet. However, $e^0 =1$ . Hence there can be no such function $f$ . On the other hand, there is a function $g$ with

$\int _0^x g(t) dt = e^x-1$

namely, $g(x) = e^x$ . This subtlety arises from the fact that an accumulation function

$F(x) = \int _a^x f(t) dt$

gives a specific antiderivative of $f$ , the one that when evaluated at $x=a$ is zero.

3 Definite integrals

A definite integral computes signed area:

$\int _a^b f(x) dx = \text {``the signed area between the $x$-axis and $f$''}$

Here we always have limits of integration, both of which are numbers. Moreover, definite integrals have definite values, the signed area between $f$ and the $x$ -axis. Pay attention to the notation:

Where $F'(x) = f(x)$ .

Definite integrals limits of integration, and they compute .

2025-01-06 19:53:17

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1 Indefinite integrals

2 Accumulation functions

3 Definite integrals

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