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.

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$\ds{\iiint\cos\pars{ax + by + cz}\,\dd x\,\dd y\,\dd z:\ {\large ?}
     \qquad\qquad\mbox{and}\qquad\qquad x^{2} + y^{2} + z^{2} \leq 1}$

Lets $\ds{\vec{k} \equiv \pars{a,b,c}}$:
  \begin{align}&\color{#c00000}{\iiint\cos\pars{ax + by + cz}\,\dd x\,\dd y\,\dd z}
=\Re\iiint\expo{\ic \vec{k}\cdot\vec{r}}\,\dd^{3}\vec{r}
=\Re\int_{0}^{1}\color{#00f}{%
\int_{\Omega_{\vec{r}}}{\expo{\ic\vec{k}\cdot\vec{r}}}
\,{\dd\Omega_{\vec{r}} \over 4\pi}}
\,4\pi r^{2}\,\dd r
\end{align}

\begin{align}&\color{#00f}{%
\int_{\Omega_{\vec{r}}}{\expo{\ic\vec{k}\cdot\vec{r}}}
\,{\dd\Omega_{\vec{r}} \over 4\pi}}
={1 \over 4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}
\expo{\ic kr\cos\pars{\theta}}\sin\pars{\theta}\,\dd\theta\,\dd\phi
={\sin\pars{kr} \over kr}
\end{align}

Then,
  \begin{align}&\color{#c00000}{\iiint\cos\pars{ax + by + cz}\,\dd x\,\dd y\,\dd z}
=\int_{0}^{1}{\sin\pars{kr} \over kr}\,4\pi r^{2}\,\dd r
={4\pi \over k^{3}}\
\overbrace{\int_{0}^{k}t\sin\pars{t}\,\dd t}
^{\ds{=\ \sin\pars{k} - k\cos\pars{k}}}
\end{align}

\begin{align}&\color{#66f}{\large
\iiint_{x^{2}\ +\ y^{2}\ +\ z^{2}\ \leq\ 1}
\cos\pars{ax + by + cz}\,\dd x\,\dd y\,\dd z}
\\[3mm]&=\color{#66f}{\large 4\pi\,
{\sin\pars{\root{a^{2} + b^{2} + c^{2}}}
- \root{a^{2} + b^{2} + c^{2}}\cos\pars{\root{a^{2} + b^{2} + c^{2}}}
\over \pars{a^{2} + b^{2} + c^{2}}^{3/2}}}
\end{align}
