Integral of $\ln(\arcsin(x))$

Question

I tried a u-substitution of $u=\arcsin(x)$ but that resulted in a bunch of bad square roots. I also looked it up on Wolfram Alpha and it has the $\text{Si}$ function but I don't really know what that is. It seems like an interesting integral so I wanted to know a bit more about how it could be solved. Anyone have any tips?

Answer 1

That function has no elementary primitive. If you do $x=\sin(t)$ and $\mathrm dx=\cos(t)\,\mathrm dt$, then you get$$\int\log(t)\cos(t)\,\mathrm dt.$$Doing integration by parts, you get that this is equal to$$\log(t)\sin(t)-\operatorname{Si}(t),$$where$$\operatorname{Si}(t)=\int_0^t\frac{\sin u}u\,\mathrm du.$$

Answer 2

As said in José Carlos Santos's answer, you cannot avoid the special function.

However, since $x$ is limited to the a quite small range, you could get a reasonable approximation composing Taylor series for the integrand $$\arcsin(x)= x+\frac{x^3}{6}+\frac{3 x^5}{40}+\frac{5 x^7}{112}+\frac{35 x^9}{1152}+\frac{63 x^{11}}{2816}+O\left(x^{13}\right)$$ $$\log(\arcsin(x))=\log (x)+\frac{x^2}{6}+\frac{11 x^4}{180}+\frac{191 x^6}{5670}+\frac{2497 x^8}{113400}+\frac{14797 x^{10}}{935550}+O\left(x^{12}\right)$$ $$\int\log(\arcsin(x))\,dx=x (\log (x)-1)+\frac{x^3}{18}+\frac{11 x^5}{900}+\frac{191 x^7}{39690}+\frac{2497 x^9}{1020600}+\frac{14797 x^{11}}{10291050}+O\left(x^{13}\right)$$ which does not seem to be too bad if we compute $$I(a)=\int_0^a\log(\arcsin(x))\,dx$$ as shown in the table below $$\left( \begin{array}{ccc} a & \text{approximation} & \text{exact} \\ 0.05 & -0.199780 & -0.199780 \\ 0.10 & -0.330203 & -0.330203 \\ 0.15 & -0.434380 & -0.434380 \\ 0.20 & -0.521439 & -0.521439 \\ 0.25 & -0.595693 & -0.595693 \\ 0.30 & -0.659661 & -0.659661 \\ 0.35 & -0.714988 & -0.714988 \\ 0.40 & -0.762827 & -0.762827 \\ 0.45 & -0.804020 & -0.804020 \\ 0.50 & -0.839204 & -0.839204 \\ 0.55 & -0.868866 & -0.868865 \\ 0.60 & -0.893380 & -0.893379 \\ 0.65 & -0.913035 & -0.913030 \\ 0.70 & -0.928039 & -0.928025 \\ 0.75 & -0.938537 & -0.938500 \\ 0.80 & -0.944604 & -0.944510 \\ 0.85 & -0.946250 & -0.946016 \\ 0.90 & -0.943407 & -0.942828 \\ 0.95 & -0.935919 & -0.934459 \\ 1.00 & -0.923525 & -0.919179 \end{array} \right)$$