a question about the evaluation of triple integral, I am stuck! How to use the method of orthogonal transformation to figure out the triple
integral ?. I am stuck about it! The triple integral is:
$$
\iiint\cos\left(ax + by + cz\right)\,{\rm d}x\,{\rm d}y\,{\rm d}z
\qquad\mbox{and}\qquad x^{2} + y^{2} + z^{2} \leq 1
$$
My solution: I want to suppose
$\quad u = ax + by + cz\,,\quad v=y\quad$ and $\quad w=z$.
\begin{align}
&\mbox{Then}\quad\iiint\cos\left(ax + by +cz\right)\,{\rm d}x\,{\rm d}y\,{\rm d}z
=\iiint { 1\over{ a}}\cos\left(u\right)\,{\rm d}u\,{\rm d}v\,{\rm d}w
\\[3mm]&\mbox{and}\quad
\left({1\over a}\,u - {b\over a}\,v - {c\over z}\,w\right)^{2} + v^{2} + w^{2}
\leq 1
\end{align}
But I don't know how to continue. Is that right ?. Or can someone use other methods to solve the question ?. You don't need to use orthogonal transformation necessarily.

I think if you change all variable to their negatives, $(x,y,z)\rightarrow (-x,-y,-z)$ the integral changes sign. Therefore $\int \int \int =0$
