Generalization of Dirichlet integral Is it correct that
$$\lim_{L\to\infty} \int_0^L d x \int_0^L dy \int_0^L d z \frac{\sin(x+y)}{x+y}\frac{\sin(y+z)}{y+z}\frac{\sin(z+x)}{z+x} =\frac {\pi^3}{16}. $$
or does anybody has a reference for this? The value $\frac{\pi^3}{16}$ comes from numerics. 
The identity
$$ \lim_{L\to\infty}\int_0^L \frac{\sin(x)}{x} = \frac \pi 2 $$
(which is sometimes also called Dirichlet integral) is well known and there are many ways to prove it. However, I need the slight generalization of this above. 
 A: A lengthy method, but it works. First, we rewrite:
$$\sin(x+y) \sin(y+z) \sin(z+x)=\frac{1}{4} \left(\sin(2x)+\sin(2y)+\sin(2z)-\sin(2(x+y+z)) \right)$$
Now let's change the variables (keeping the same letters for simplicity):
$$x \to \frac{1}{2}x^2, \qquad y \to \frac{1}{2}y^2, \qquad z \to \frac{1}{2}z^2$$
$$I= 2 \int_0^\infty \int_0^\infty \int_0^\infty \frac{x y z dx dy dz}{(x^2+y^2)(y^2+z^2)(z^2+x^2)} \left(\sin(x^2)+\sin(y^2)+\sin(z^2)-\sin(x^2+y^2+z^2) \right)$$
Now we introduce spherical coordinates:
$$x= r \sin \theta \cos \phi, \qquad y = r \sin \theta \sin \phi, \qquad z= r \cos \theta$$
The limits are obvious:
$$I= 2 \int_0^\infty \int_0^{\pi/2} \int_0^{\pi/2} \frac{r^5 \sin^3 \theta \cos \theta \sin \phi \cos \phi dr d \theta d \phi}{r^6 \sin^2 \theta (\sin^2 \theta \sin^2 \phi+\cos^2 \theta)(\sin^2 \theta \cos^2 \phi+\cos^2 \theta)} \times \\ \times \left(\sin(r^2 \sin^2 \theta \cos^2 \phi)+\sin(r^2 \sin^2 \theta \sin^2 \phi)+\sin(r^2 \cos^2 \theta)-\sin(r^2) \right)$$
Simplifying and using:
$$\int_0^\infty \frac{\sin(a^2 r^2) dr}{r}=\frac{\pi}{4}$$
We obtain:
$$I= \pi \int_0^{\pi/2} \int_0^{\pi/2} \frac{\sin \theta \cos \theta \sin \phi \cos \phi d \theta d \phi}{(\sin^2 \theta \sin^2 \phi+\cos^2 \theta)(\sin^2 \theta \cos^2 \phi+\cos^2 \theta)}=\pi \int_0^{\pi/2} \int_0^{\pi/2} \frac{\sin 2 \theta \sin 2 \phi d \theta d \phi}{4 \cos^2 \theta+\sin^2 2 \phi \sin^4 \theta}$$
Changing the variables:
$$2 \theta \to \theta, \qquad 2 \phi \to \phi$$
$$I= \pi \int_0^{\pi} \int_0^{\pi} \frac{\sin \theta \sin \phi d \theta d \phi}{8 (1+\cos \theta)+\sin^2 \phi (1-\cos \theta)^2}$$
Substituting:
$$u= \cos \theta, \qquad v = \cos \phi$$
$$I= \pi \int_{-1}^1 \int_{-1}^1 \frac{d u d v}{8 (1+u)+(1-v^2) (1-u)^2}$$
Substituting:
$$1-u= t$$
$$I= \pi \int_0^2 \int_{-1}^1 \frac{d v d t}{(t-4)^2-t^2 v^2}=\pi \int_0^2 \int_{-1}^1 \frac{d v d t}{(t-4-tv)(t-4+tv)}$$
Taking the elementary integral w.r.t. $v$, we have:
$$I= \pi \int_0^2 \frac{dt}{t(t-4)} \ln \left(1-\frac{t}{2} \right)$$
Substituting:
$$t= 2s$$
$$I= \frac{\pi}{2} \int_0^1 \frac{ \ln (1-s) ds}{s(s-2)}=\frac{\pi}{4} \left( \int_0^1 \frac{ \ln (1-s) ds}{s-2} -\int_0^1 \frac{ \ln (1-s) ds}{s}\right)$$

$$I=\frac{\pi}{4} \left( \frac{\pi^2}{12}+\frac{\pi^2}{6}\right)=\frac{\pi^3}{16}$$

The last two logarithmic integrals are well known, and their values can be determined by various methods, for example the series expansion.
A: As David C. Ullrich did, let
$$ u=x+y,v=y+z,w=x+z. $$
Then
$$ x=\frac{1}{2} (u-v+w),y=\frac{1}{2} (u+v-w),z=\frac{1}{2}(-u+v+w) $$
and
$$ \frac{\partial(x,y,z)}{\partial (u,v,w)}=\frac12.$$
Also note that for $0\le x,y,z\le L$, $0\le u,v,w\le 2L$. So
\begin{eqnarray}
&&\lim_{L\to\infty} \int_0^L d x \int_0^L dy \int_0^L d z \frac{\sin(x+y)}{x+y}\frac{\sin(y+z)}{y+z}\frac{\sin(z+x)}{z+x} \\
&=&\lim_{L\to\infty} \int_0^{2L} d u \int_0^{2L} dv \int_0^{2L} d w \frac{\partial(x,y,z)}{\partial (u,v,w)}\frac{\sin(u)}{u}\frac{\sin(v)}{v}\frac{\sin(w)}{w} \\
&=&\frac12\lim_{L\to\infty}\bigg[\int_0^{2L} d u \frac{\sin(u)}{u}\bigg]^3\\
&=&\frac {\pi^3}{16}.
\end{eqnarray}
