Problem: Assume $ a^2 + b^2 + c^2 = 1 $. Calculate the improper integral $\int_{B(\mathbf{0}, 1)} \frac{\mathrm{d} x \mathrm{~d} y \mathrm{~d} z}{1-a x-b y-c z}$ where $B(\mathbf{0}, 1)=\left\{x^2+y^2+z^2 \leq 1\right\}$ is the unit ball in $ \mathbb{R^3}$.
Attempt: Assuming $ a,b,c \neq 0 $ ( I had difficulty with the simpler cases as well where for example $ a=1 , b=c=0$ ) I performed changed of variables $ u = 1-ax-by-cz , v = by , w = cz \iff x = \frac{1-u-v-w}{a} , y =v/b , z = w/b $ ,
the absolute value of the jacobian will be $ \frac{1}{abc} $ and I get that the integrand will be $ \frac{1}{abc} \cdot \frac{1}{u} $, the problem is, I'm having difficulty determining the new set under integration according to the diffeomorphism ( induced by the change of variables ) hence I can't proceed to calculate the integral. I know the new set of integration will have
$ (\frac{1-u-v-w}{a})^2 + (v/b)^2 + (w/c)^2 \leq 1 $
but I don't know how to continue and hopefully, to use Fubini's theorem.
Any ideas?
Thanks for the help!
As the domain as well as the measure of the integral $$I(\vec{n})=\!\!\!\int\limits_{B(0, 1)}\! \! d^3x \; \frac{1}{1- \vec{n}\cdot \vec{x}}, \qquad |\vec{n}|=1,$$ are invariant under rotations, we have $$I(R \,\vec{n})=I(\vec{n}) \quad \forall \,R \in {\rm SO(3)},$$ i.e. the result is independent of the direction of the unit vector $\vec{n}=(a,b,c)^T$. Using spherical coordinates $$\begin{align} x_1=r \, \sin \theta \, \cos \varphi, \quad x_2=r \,\sin \theta \,\sin \varphi, \quad x_3= r \, \cos \theta \end{align}$$ and choosing $n_1=n_2=0, \; n_3=1$, we obtain (with $u= \cos \theta$) $$\begin{align} I(\vec{n})&= \int\limits_0^1 \!dr \,r^2 \int\limits_{-1}^1 \! du \int\limits_0^{2\pi}\! d\varphi \;\frac{1}{1-r \,u}\\[5pt] &=2\pi\int\limits_0^1 \! dr \, r^2 \int\limits_{-1}^1 \! \frac{du}{1-r \, u}\\[5pt] &=2\pi \int\limits_0^1\! dr \, r \, \log\frac{1+r}{1-r} \\[5pt]&=2\pi. \end{align}$$