Compute $$\iiint_E \frac{ \,dx \,dy \,dz}{\sqrt{x^2+ y^2 +(z-2)^2}}$$
where $E$, i.e. the domain of integration, is specified by $$x^2+y^2+z^2 \le 1$$
I tried using spherical coordinates
But i end up getting an integral solving which is an onerous task
Can anyone suggest an alternatively easier method ?
$$∫∫∫ \dfrac{dxdydz}{\sqrt{x^2 + y^2 + (z-2)^2}}{=∫∫∫ \dfrac{R^2\sin\theta d\theta d\phi dR}{\sqrt{R^2+4-4R\cos\theta}}\\=2\pi\int_{0}^{1}\int_{\theta=0}^{\theta=\pi}\dfrac{R^2 d(-\cos\theta) dR}{\sqrt{R^2+4-4R\cos\theta}}\\=2\pi\int_{0}^{1}\dfrac{R}{2}\sqrt{R^2+4+4Ru}|_{-1}^{1}dR=2\pi\int_{0}^{1}R^2dR\\=\dfrac{2\pi}{3}}$$