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\end{equation}  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{array}{|c|c|}  Eqn & d/d1 \\  \hline  \int_0^\pi sin(x)dx =2 & \int_0^\pi xcos(x)dx = -2  \hline  \end{array}  \end{equation}