How do you denote the objects of differentiation in double integral?

Question

I do not know if there is standard notation for (partial) differential in evaluation of double integral, so I figure out a strange notation. For example, when evaluate the following double integral \begin{gather*} \int_0^{\frac{\pi}{2}}\int_0^2x^2y\cos(xy^2) d y d x, \end{gather*} since the inner integral is to integrate with respect to the variable $y,$ while $x$ is just viewed as a temporary parameter. I would write the calculation of this double integral like this: \begin{align*} \int_0^{\frac{\pi}{2}}\int_0^2x^2y\cos(xy^2) d y d x=\int_0^{\frac{\pi}{2}}\int_0^2 \frac{x^2}{2} \cos(xy^2)d_y(y^2) dx, \end{align*} where the subscripted notation $d_y$ of differentiation $``d"$ indicates that we are to differentiate with respect to $y.$ Thus, $d_y(xy^2)=\frac{xy}{2}d y,$ and so, \begin{align*} &\int_0^{\frac{\pi}{2}}\int_0^2x^2y\cos(xy^2) d y d x=\int_0^{\frac{\pi}{2}}\int_0^2 \frac{x^2}{2} \cos(xy^2)d_y(y^2) dx\\ =&\int_0^{\frac{\pi}{2}}\int_0^2\frac{x}{2}\cos(xy^2)d_y(xy^2)d x=\int_0^{\frac{\pi}{2}}\frac{x}{2}\sin(xy^2)\bigg|_{y=0}^{y=2}dx. \end{align*} My question is, is there standard notation for the differential of xy^2, with respect to $y,$ not to $x?$

Answer 1

I don't know about standard notation for this. Usually, when integrating something like $x^2y\cos(xy^2)dy$, I know that the result is $\sin(xy^2)$ times some constant, so I write it and then calculate the derivative in head to figure out the appropriate constant. If it is not so easy, I just write the substitution in full: $z = xy^2$, $dz = 2xydy$.

Added: Of course you can write $d_y z = 2xydy$ to stress the fact that you consider $z$ as a function of only $y$. But you can do this also by writing $z(y) = xy^2$ in the first place.