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 \(F'(x) = f(x)\).

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?
Devyn is correct Riley is correct Both students are correct Neither student is correct

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 \(F'(x) = f(x)\).

True or false: There exists a function \(f\) such that
\[ \int _0^x f(t) dt = e^x \]
true false

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 \(F'(x) = f(x)\).

2025-01-06 19:53:17