# What is $dx$ on an integral? [duplicate]

I've heard from some of my teachers it's a bilineal form and some other stuff, nobody actually ever explained me the reason of it. Of course i've done practical problems in which $dx$ is a "very small part of somethign", but what can we say about it in some "higher grade" notes?

$dx$ is called a differential and you can think of it as the "operator" $d$, known as the differential operator, acting on the expression $x$.

If you have seen the notation

$$y'(x)=\frac{dy}{dx},$$ the same operator is in use, and you are allowed to take the ratio of two differentials.

In the past, $dx$ was understood as an infinitesimally small quantity. In more modern notation, $dx$ represents the linear part of the variation of the expression that follows it.

For example the variation around $x^2$ is $(x+h)^2-x^2=2xh+h^2$, of which the linear part is $2xh$ (when $h$ varies, $2xh$ varies proportionally, while $h^2$ varies quadratically). For this reason, we denote $dx^2=2x\ dx$.

The integration operator $\int$ can be seen as the inverse of $d$, such that

$$\int y'(x)\,dx=\int dy=y(x)+C.$$

It solves the problem "what is the function of which the differential is what follows").

Hence

$$\int 2x\,dx=\int dx^2=x^2+C.$$ The additive constant $C$ is there because the inverse is indeterminate to a constant.

As clumsy as is may appear, this notation becomes indispensable for functions of several variables, where you need to specify on which variable you differentiate/integrate.

dx on an integral comes (at least) from the definition of Stieltjes integral. Here the definition for real functions: Let $f, x:[a,b]\longrightarrow\mathbb{R}$ be real functions defined on the interval $[a,b]$. Consider a partition of the interval $[a,b]$, $P^{*} = (P,\xi)$, where: $$P = \lbrace a = t_{0} < t_{1} < \ldots < t_{k}=b\rbrace, \ \ t_{i-1}\leqslant \xi_{i} \leqslant t_{i}.$$

The Stieltjes sum associated to this partition and the functions $f$ and $x$ is defined by:

$$\Sigma(f,x,P {*}) = \Sigma(P {*}) = \sum_{i=1}^{k}f(\xi_{i})[x(t_{i}) - x(t_{i-1})].$$ Then, (here is the answer), the Stieltjes integral of $f$ in relation to $x$ is given by: $$\int_{a}^{b}f(t)dx = \displaystyle \lim_{\vert x\vert\rightarrow 0}\sum(P^{*}).$$ Observe that, the left rand of the equation is just a notation.

Here "very small" means "infinitesimal". The problem is that the real number system has the disadvantage of not containing any nonzero infinitesimals. However, the real number system has an extension or an enlargement called the hyperreal number system. The hyperreal view helps clarify the meaning of $f(x)dx$ in an integral $\int_a^b f(x)dx$.

From the hyperreal viewpoint, the integral is defined as the standard part of an infinite Riemann sum of areas of infinitely thin strips: $\sum_i f(x_i)\Delta x$. In the Riemann sum, the term $f(x_i)\Delta x$ is literally a product, and therefore commutative. Since the infinite Riemann sum uniquely determines the integral (via applying standard part), it follows that in the integral also we are free to commute $f(x)$ and $dx$.