I am currently trying to get a deeper understanding of multivariable calculus and I am finding my one stumbling block to be the variable of integration. In single variable calculus, I have always considered the variable of integration (in antiderivatives) as the "what do I have to differentiate the answer with respect to in order to get the integrand." For instance, in the following:
$$\int 2x dx=x^2$$
I have always interpreted the dx as meaning I have to differentiate my answer ($x^2$) with respect to x in order to get my integrand ($2x$). In other words, I have to take $\frac{d}{dx}$ of $x^2$ to get $2x$.
I am struggling to understand a certain concept though: multiplication of variables of integration. Say, for instance, that I have the following:
$$\int (x -6)^2 dx$$
If I define $u = x - 6$, I can write the following equivalent integral
$$\int u^2 \frac{du}{dx}dx$$
It seems to me that now there are "multiple" variables of integration. Am I looking for an answer that I have to differentiate with respect to $u$ or $x$ in order to get the integrand? This can be simplified to:
$$\int u^2 du$$
Now, it is clear that I am looking for a function that I have to differentiate with respect to $u$ in order to get my integrand. However, I don't understand why the variables of integration ($dx$) can just cancel. My understanding is that they must mean more than "what do I have to differentiate my answer...." In the case of definite integrals, it is relatively clear (I can imagine them as small changes along a certain axis). But, with indefinite integrals, I cannot think of what they represent. Any help would be very much appreciated. Thank you!