TITLE: Why can we treat infinitesimals as real numbers in integration by substitution?
QUESTION [7 upvotes]: During integration by substitution we normally treat infinitesimals as real numbers, though I have been made aware that they are not real numbers but merely symbolic, and yet we still can, apparently, treat them as real numbers. For instance, consider we want to integrate the expression $3x(x^4+1)^3$. A common way to do this is to let $u=x^4+1$, where $\frac{du}{dx}=4x^3$, and thus $du=4x^3dx$ which is appropriately used in our substitution to obtain $\int3x(u)^4 du$, and then we simply directly integrate this new integrand. However, while I understand the process and why we do it in such a manner, I am perplexed as to why we can still rigorously treat the infinitesimals as real numbers. So, my question is if anyone can elaborate on exactly why it is logically rigorous to treat infinitesimals as real numbers during substitution for integration. 
(Note: My question does not concern as to what "dx" means in integration simply because my question is defined in the prospect of treating infinitesimal derivatives as ratios specifically in integration by substitution, where other questions do not specifically address. )

REPLY [3 votes]: The justification of substitution method for finding indefinite integrals is the chain rule of derivative. Suppose $F(x)$ is anti-derivative of $f(x)$ so that $$F(x) = \int f(x)\,dx$$ and $F'(x) = f(x)$. Now if we put $x = g(t)$ then and apply chain rule we get $$\frac{d}{dt}\{F(g(t))\} = F'(g(t))g'(t) = f(g(t))g'(t)$$ and hence we get $$F(g(t)) = \int f(g(t))g'(t)\,dt$$ or the rule of substitution for indefinite integration.
As you can see that it has nothing to do with infinitesimals or differentials. In order to memorize this rule we sometimes give informal version: Let $x = g(t)$ so that $dx = g'(t)\,dt$ and hence $$\int f(x)\, dx = \int f(g(t))g'(t)\,dt$$ So this kind of language is used only to help remember the rule.
It is better not to give the $dx$ in both $d/dx$ and $\int\,dx$ notation a proper meaning. They are best suited for aid in remembering rules of calculus. Although it is possible to give a formal meaning to these symbols (what we call differentials) but it does not lead to any concept which is more useful than derivatives or integrals.