Indefinite Integral

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We introduce the idea of the indefinite integral - all the antiderivatives!

We’ve discussed the idea of the antiderivative, and now we aim to introduce the area where antiderivatives are used most often - the indefinite integral.

Video Lecture

Text with Additional Details

The indefinite integral is a notation used to represent the class of antiderivatives. What exactly we mean by “class” will be covered in another segment, but in this segment we aim to introduce the notation, first established by Leibnitz. In particular, Leibnitz introduced an elongated German “S” to represent the integral.In particular, for a function $f(x)$ we would write the indefinite integral of $f(x)$ as:

$\int f(x) \dx$

This is an impressively compact bit of notation which we will explain part by part.

The integral sign $\int$ represents the fact that we are doing an indefinite integral - meaning that we want the (class of) antiderivative(s) of $f(x)$ . The $dx$ at the end is easy to overlook but it turns out to be very important. It is the variable which is being “integrated against” - in other words, it is the variable whose derivative process we are trying to reverse. You should think of the integration notation kind of like parentheses, you have the “starter” symbol “ $\int$ ” which is like the opening of the parenthesis $($ , and then you have the “ending” symbol “ $dx$ ”, which acts like the closing parenthesis $)$ . Finally, the “ $f(x)$ ” is the function which we are trying to find an antiderivative for - referred to as the integrand.

Integrand The function inside the integral notation is known as the integrand. Specifically, in the expression $\int f(x) \dx$ the function $f(x)$ would be called the integrand.

1 : Which of the following are true (select all that apply)

Every integral must have both $\int$ and a “closing” expression denoting the vraiable of interest, like dx, dy, or du. Every integral should have both an $\int$ and something like $\dx$ , but it isn’t necessary. Every integral must have an integrand, even if it may not be obvious. Integration notation, like the $\int$ symbol is just archaic nonsense that we won’t really use in the future.

Consider, by way of example, $F(x) = x^3$ . Then $F'(x) = 3x^2$ . So, if we wanted to find the (class of) antiderivative(s) for $3x^2$ we would write;

$\int 3x^2 \dx$

To compute this we would reverse the process to get back to $F(x)$ . There is one additional caveat, which we haven’t necessarily discussed yet but we will shortly, and that is that we need to include a $+C$ to our result. Again, the “why” here will be explained (as well as what this “ $+C$ ” nonsense represents) but for now just roll with this part. So we have:

$\int 3x^2 \dx = x^3 + C$