I have a function $f(r)$ in polar coordinates, for some positive $a$ and $b$, with $0<n<1$, defined on the unit disk ($0<r<1$) $$ f(r) = a + \dfrac{b}{(1-r)^n}$$

I would like to integrate this function on a slice of the unit disk of width $s$.

I think this corresponds to $$\int_{\theta=-\arccos(1-s)}^{\arccos(1-s)} \int_{r=(1-s)\sin\theta}^1f(r)r\ \mathrm{d}r\ \mathrm{d}\theta$$

The inner integral is doable enough. $$F(r) = \frac{a}{2}r^2 + \frac{b(r-1)(r-nr+1)}{(1-r)^n(n^2-3n+2)}$$

However, when filling in the limits, we get quite the draconian equation, and integrating that is well beyond my capabilities. I tried integrating in Cartesian coordinates, but the result was not much better. (note: I do not know if this is actually possible. Ideas on how to approximate this for small $s$ are welcome too).

Is there a more straightforward method suited for integrating this particular geometry and/or equation?


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