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\end{equation}  Where $c$ is the integral from $a$ to $b$ of $u$. That way if the $xu$ term vanishes, the identity holds... Thus the operation $d/d\mathbb{I}$ was mapping the integral \begin{equation}  \frac{d/d\mathhbb{I}}\int_a^b f(x) dx \to \int_a^b xf'(x) dx  \end{equation}  and the answer \begin{equation}  \frac{d/d\mathhbb{I}}I \to -I + [xf(x)]_a^b  \end{equation}