When integrating sin(x/2) for example using substitution as in:
$\int \sin(\frac x2)$ $dx$
$\frac x2 = u $,
$\frac{du}{dx} = \frac{1}{2} \Rightarrow 2du = dx $
$\int \sin(u)$ $2du$
etc
How is it that the du can be multiplied by 2 here, I thought that the dx represented a 'with respect to x' or a very small value of x. So does this mean that $ 2du = dx $ means 2 * a really small change in u is a really small change in x? And how come it can be separated from the main part of the integral so you just multiply by two at the end with the du 'disappearing'?
$-\cos(u) * 2$
$-2\cos(\frac{x}{2})$
Best Answer
This is a good question.
In general we take
$$ \int f(x)\; dx $$ as a symbol on its own. In general we can't break down the symbol and assignment meaning to each of the component. So $\int$ isn't a defined quantity. (Of course $f(x)$ makes sense). Likewise, $dx$ on its own isn't defined.
Now, we do have integration by substitution. According to this, as a matter or pure notation we allow for writing $dx$ on its own. It is a result that when we do that, like you have done in your example, then it actually "works".
So we are allowed to treat $du$ and $dx$ as quantities that are defined, and we are allowed to multiply and divide by them when we do integration by substitution. So if $$ \frac{du}{dx} = \frac{1}{2} $$ we move ("multiply") by $dx$ on both sides to get $$ du = \frac{1}{2}dx. $$ Then we multiply by $2$ on both sides and get $$ 2du = dx. $$ All this means is that you are allowed to replace the $dx$ in the orignial integral by $2du$. And so you get $$ \int \sin(u)\; du. $$ So in this sense we don't in general consider $dx$ and $du$ to have a life outside of the use in integration by parts. (Ok this is not true. There are other places where we write $dx$ on its own).
In all this we don't assign an precise meaning to $dx$. So $dx$ doesn't mean "a small change". Sure we can think about it as such, but to be precise we would have to define what "small change" means.