Is it correct to write $\int_a^x f(x) dx$?

Question

The question pretty much sums it all. A few days ago when studying how to find the real part of a function knowing the imaginary part (or vice versa) I was given this formula: $$u(x, y) =\int_{x_0}^{x} \frac{\partial u} {\partial x} (x, y_0) \,dx + \int_{y_0}^{y} \frac{\partial u} {\partial y} (x, y)\, dy$$ Is this a correct notation? If not, is there a simple proof to convince someone that this is an incorrect notation? So $\int_a^x f(x) \,dx$ or $\int_a^x f(t)\, dt$?

Answer 1

Under the usual conventions for variable binding operators the notations $\int_c^xf(x)dx$ and $\int_c^xf(y)dy$ are both correct and mean the same thing. However, if you want to be kind to your readers, I'd suggest you avoid using $x$ as a free variable and a bound variable in one formula.

Answer 2

I would say it is not correct as you are using the same letter for variable of integration and one of the extremes of integration. I would probably write: $$\int_{x_0}^x \frac{\partial u}{\partial x^1}(x^1,x^2_0)\,dx^1+\int_{y_0}^y \frac{\partial u}{\partial x^2}(x^1,x^2)\,dx^2.$$

Answer 3

It is quite incorrect, because the $x$ in the integrand is a dummy variable, i.e. is does not appear in the result, just like when you write $$\sum_{k=1}^5 k^2,$$ $k$ does not appear in the result, since the closed form of this sum is $55$. Here $k$ is just a temporary variable which lets you know at which step of the computation you've arrived. After the computation is finished, you throw it awway. The same is true for integrals.

Using $x$ in both places reminds me of Groucho Marx's sentence: ‘I don't want to belong to any club that will accept me as a member.’