Prove that $$\iint\limits_S f(ax+by+cz)\ ds=2\pi \int_{-1}^{1} f(u\sqrt{a^2+b^2+c^2})\ du,$$ where $f(u) \in C^{(1)}$, and $S$ is $x^2+y^2+z^2=1$.
My attempt. Let $u=\dfrac{ax+by+cz}{\sqrt{a^2+b^2+c^2}}$. Then $f(u\sqrt{a^2+b^2+c^2})= f(ax+by+cz)$. But I don’t know what should I do next.