# Solve $\int_0^1 \int_0^{2\pi}\frac{ax-x^2\sin(\theta)}{\sqrt{a^2-2ax\sin(\theta)+x^2}}d\theta dx$

Solve $$\int_0^1 \int_0^{2\pi}\frac{ax-x^2\sin(\theta)}{\sqrt{a^2-2ax\sin(\theta)+x^2}}d\theta dx$$ This integral is from the following paper : Frictional coupling between sliding and spinning motion and can be understood as an integration of unit vector field parallel to the velocity of a spinning disk which is sliding with the speed of $a$ over a unit disk. The evaluation of the integral is claimed to be

$$\frac{4}{3}\frac{(a^2+1)E(a)+(a^2-1)K(a)}{a}$$ for $a<1$, and

$$\frac{4}{3}((a^2+1)E(1/a)+(a^2-1)K(1/a))$$ for $a\geq1$ in the paper, where $K$ and $E$ are the first and second kind of complete elliptic integrals respectively. But for me this integration is not clear since if I do the integration for $x$ first then it results in a term with natural log which is irrelevant to elliptic integral, and doing it for $\theta$ first results in the terms including $E(\frac{4ax}{(a-x)^2})$ and $K(\frac{4ax}{(a-x)^2})$ which seems hard to be reduced to $E(a)$ and $K(a)$. What approach would give the most clarified evaluation? Any advice or help will be appreciated.

• strange...Have u checked there result numerically? – tired Feb 1 '16 at 11:05
• a small addition: i doubt that u can exchange integration limits freely here because the integral is singular for some choices of parameters (for example $\Theta=0, a<0$) and one has to be a little bit carefule – tired Feb 1 '16 at 11:39
• I would try the change of variables $x\sin\theta=aw$ and $x\cos\theta=at$. – Pierpaolo Vivo Feb 1 '16 at 13:16
• a quick check in mathematica suggests that the results they obtained are incorrect. For example setting $a=1.5$ i obtain $2.95567$ using NIntegrate and a value of $11.1862$ using the result they give. values of $a$ lead to similar descripancies – tired Feb 1 '16 at 14:42
• @tired Thank you for you check. But I think the claimed evaluation value should not exceed $\pi$ for any value of $a$. (There is a graph for it in the paper above) – generic properties Feb 1 '16 at 15:27