I just saw the following notation of an integral on a paper:
$$ F(t)=\int^{t}_{0}{\frac{d\tau}{(t-\tau)^{1/2}}}\left[\frac{d\Delta}{d\tau}-\Delta^3\right] $$
where $\tau$ is that 'dummy' variable from the convolution theorem. My first question is: is it normal or usual to write the integrated after the integration domain (i.e., after $d\tau$)? It seems that I'm missing something and I've never seen this notation, usually, the integrated comes more like 'inside' the integral. The author sometimes put it inside and others outside.
My second question is, suppose that I have the linear function $\Delta=At$, how can I evaluate its derivative with respect to $\tau$? In practice it doesn't look like $0$ to me.
Finally, if I was going to solve a function similar to this one numerically, the best option I have is methods like Simpson's rule? I can't see how to integrate over $\tau$ to get an answer as a function of $t$ in those cases. Any literature provided will be very helpful as well.
I really appreciate any help you can provide.