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In theory for any function which ends up as a polynomial, the differentiation and multiplication with $x$ should leave the form unchanged except for a numeric factor, which will drop out in some circumstances.  A simpler set of examples with the normal integral on the left, and the answer on the right\begin{equation}  \begin{equation}  \begin{array}{|c|c|}  Eqn & d/d1 \\  \hline  Eqn & d/d\mathbb{I} \\  \hline  \int_0^\pi sin(x)dx =2 & \int_0^\pi xcos(x)dx = -2 \\  \hline   \end{array}  \end{equation}  Contradictory examples to this are  \begin{equation}  \begin{array}{|c|c|}  \hline  Eqn & d/d\mathbb{I} \\  \hline   \int_0^\pi cos(x)dx =0 & \int_0^\pi xsin(x)dx = \pi \\  \int_0^\pi x^2 dx =\pi^3/3 & \int_0^\pi 2x^2dx = 2\pi^3/3 \\  \hline  \end{array}  \end{equation}