I am just reading book "University physics with modern physics 14-th edition (Young & Fredman)". And on page 702 there is an example 21.9 which says:
Charge $Q$ is uniformly distributed around a conducting ring of radius $a$. Find the electric field at point $P$ on the ring axis at a distance $x$ from center.
So author first states that linear charge density $\lambda = Q/2\pi a \rightarrow \lambda = \text{d}Q/\text{d}s$ where $\text{d}s$ is a diferential of the ring length. It is also immediately clear that there won't be any net electric field in the $y$ direction as $\text{d}{E}_y$ cancel out while on the other hand all $\text{d}{E}_x$ sum up. Therefore we can write this in scalar form:
\begin{equation*} \begin{split} \text{d}E_x &= \text{d}E \cdot cos(\alpha)\\ \text{d}E_x &= \frac{1}{4\pi\varepsilon_0}\frac{\text{d}Q}{r^2} \cdot cos(\alpha)\\ \text{d}E_x &= \frac{1}{4\pi\varepsilon_0}\frac{\text{d}Q}{r^2} \cdot \frac{x}{r}\\ \text{d}E_x &= \frac{1}{4\pi\varepsilon_0}\frac{\text{d}Q}{(x^2+a^2)^2} \cdot \frac{x}{x^2 + a^2}\\ \text{d}E_x &= \frac{1}{4\pi\varepsilon_0}\frac{x}{(x^2 + a^2)^{3/2}}\cdot \text{d}Q\\ \text{d}E_x &= \frac{1}{4\pi\varepsilon_0}\frac{x\cdot \lambda}{(x^2 + a^2)^{3/2}}\cdot \text{d}s\\ \end{split} \end{equation*}
This is all fine, but then he integrates over all the ring's length. But if we have equation we have to integrate in a same way on both sides right? So I think integration should look like this:
\begin{equation*} \begin{split} \int_0^{2\pi a}\text{d}E_x\, \text{d}s &= \int_0^{2\pi a}\frac{1}{4\pi\varepsilon_0}\frac{x\cdot \lambda}{(x^2 + a^2)^{3/2}}\cdot \text{d}s \, \text{d}s\\ \int_0^{2\pi a}\text{d}E_x\, \text{d}s &= \frac{1}{4\pi\varepsilon_0}\frac{x\cdot \lambda}{(x^2 + a^2)^{3/2}}\cdot \int_0^{2\pi a} \text{d}s \, \text{d}s \end{split} \end{equation*}
What is weird to me is integral on the right. Well author of the book doesn't even integrate in a same way on both sides of equation. What he writes down is:
\begin{equation*} \begin{split} \int \text{d}E_x &= \frac{1}{4\pi\varepsilon_0}\frac{x\cdot \lambda}{(x^2 + a^2)^{3/2}}\cdot \int_0^{2\pi a} \text{d}s\\ E_x &= \frac{1}{4\pi\varepsilon_0}\frac{x\cdot \lambda}{(x^2 + a^2)^{3/2}}\cdot 2\pi a\\ \end{split} \end{equation*}
Is he alowed to do that? Why?