On Wikipedia articles, I often see $$\frac{dx}{x}$$ where the $\frac{1}{x}$ can often be absorbed into powers of $x$ in the integrand.
For example, the Mellin transform is expressed as $$\int_0^\infty x^s f(x) \frac{dx}{x}$$ instead of $$\int_0^\infty x^{s-1} f(x) \, dx.$$
What is the reason behind?
It is about the Haar measure on the multiplicative group $\left((0,\infty),\dfrac{dx}{x}\right)$. And when you are dealing with convolution in this group, one writes that $f\ast g(x)=\displaystyle\int_{0}^{\infty}g(y^{-1}x)f(y)\dfrac{dy}{y}$.