Question

This has always been something that I have always had trouble with is is integrating (dx(dy/dx)) the same as integrating (dx)? I see this a lot and it always throws me off

Answer 1

\[\Large \int\limits \frac{dy}{dx}dx\quad=\quad \int\limits f'(x)dx\quad=\quad f(x)+c\]
\[\Large \int\limits dx \quad=\quad c\]

Answer 2

Is that what you're asking about maybe? :o

Answer 3

Woops I'm so silly...

Answer 4

\[\Large \int\limits\limits dx \quad=\quad x+c\]

Answer 5

I believe so I always see something like \[\int\limits_{0}^{c}(C _{f}\frac{ dy }{ dx })dx\]
and my first instinct is to cancel out the dx so that I get something like this
\[\int\limits_{0}^{c}C _{f}dy\]
If tis is not correct then why?

Answer 6

You can't divide or multiply differentials the same way you can with normal values.
They are essentially zero quantities, so it doesn't work in the same way, you end up dividing by "zero" kind of. :P

There is a process that allows us to do things like that sometimes, move differentials around and rewrite them in other ways, like with the chain rule:
\[\Large \frac{d}{dt}y(x)\quad=\quad \frac{dy}{dt}\quad=\quad \frac{dy}{dx}\cdot\frac{dx}{dt}\]
It looks like we multiplied and divided by dx, but that's not actually what is happening.

I dunno, I can't think of the best explanation for this :( I don't have a great understanding of that...

Answer 7

Well this definitely helps thanks!