(On pause)
I have
$$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$
which looks a bit like the continuous version of Poisson distribution in $k=x$ with parameter $\lambda=d$ (or a function of $d$) such that I can write
$$ f\left(x\right)\approx\frac{\lambda^x\left(d\right) e^{-\lambda \left( d \right)}}{x!} $$
where $\lambda\left(d\right)$ is some function I need to find (and $x!$ is the gamma function since $x$ is continuous). I've tried the usual methods, but can't get very far. Is there a way of finding $\lambda\left(d\right)$, perhaps just for $d\approx2$ (which I can use if necessary).
See this screenshot, containing a plot of $f(x)$ for $d=2$
