It seems like you may have missed the beginning lectures on integration and the Fundamental Theorem of Calculus. If you have to ask what dx means then you aren't ready to move on. Go back and review the basics of integration.
Lecture 15: Antiderivatives | Video Lectures | Single Variable Calculus | Mathematics | MIT OpenCourseWare
I am going through the video above. There is some notation being introduced:
It is my understanding right now that "*dx" is only there because it is denoting sin(x) as being a derivative of some equation and that it does not at all mean sin(x) times dx. Am I wrong on that?
Also it is introduced that:
So I have studied I'm not sure yet if this is related however I find it very confusing that in the case above dx is on the top part of a fraction? I'm not sure what to make out of that.
Thanks for any responses...
That's not really so. Lecture 15 is the first lecture on the integral. Again I'm not asking what dx is I am asking if in the case of the integral notation if it means sin(x) times dx or if its just denoting sin(x) as the derivative of some unknown equation. So I read the required reading materials out of my textbook although I don't really understand everything yet. Also I'm asking why does the notation change to .
In the "first lecture on the I integral" have they not yet defined the integral?
Perhaps the "first lecture" is intended to be only introductory. There are, in fact two ways to define the integral- one is as the anti-derivative: the anti-derivative of f(x) is the "general" (That is really a "family" of functions- it will involve and added undetermined constant) function, F, such that F'= f. The other is in terms of the "Riemann sum" which you can think of a limit of a summation with each term of the form . It is, essentially, the " that becomes the "dx".
The fact that those two definitions are equivalent is the "Fundamental Theorem of Calculus". It the recognization of that that make Newton and Leibniz the founders of Calculus rather than, say, Fermat.
dx always refers to an infinitesimal interval on x.
When paired with an integral sign it stands for the continuous sum of the integrand times this infinitesimal bit of x.
It does so happen, due to the fundamental theorem of calculus, that there is an antiderivative associated with your integrand and that that antiderivative is an equivalent functional form of the integral.
As to your specific question I'm not sure I understand the confusion. What's wrong with dx/x ?
It's just an infinitesimal bit of x divided by x. Paired with an integral sign it means the integral of 1/x.