Looks like the integral is supposed to be
[tex]\displaystyle\iiint_{\mathcal W}\frac{\mathrm dx\,\mathrm dy\,\mathrm dz}{(x^2+y^2+z^2)^{3/2}}[/tex]
but it's not clear what spheres bound the region [tex]\mathcal W[/tex]. I'll assume it's the space between two spheres, both centered at the origin, with radii [tex]R[/tex] and [tex]r[/tex] such that [tex]0<r<R[/tex].
Convert to spherical coordinates, setting
[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\theta\end{cases}[/tex]
Then the region [tex]\mathcal W[/tex] is parameterized by varying over [tex]r\le\rho\le R[/tex], [tex]0\le\theta\le2\pi[/tex], and [tex]0\le\varphi\le\pi[/tex]. Note that [tex]x^2+y^2+z^2=\rho^2[/tex]. We also have the Jacobian
[tex]\mathbf J=\dfrac{\partial(x,y,z)}{\partial(\rho,\theta,\varphi)}=\begin{bmatrix}\cos\theta\sin\varphi&-\rho\sin\theta\sin\varphi&\rho\cos\theta\cos\varphi\\\sin\theta\sin\varphi&\rho\cos\theta\sin\varphi&\rho\sin\theta\cos\varphi\\\cos\varphi&0&-\rho\sin\varphi\end{bmatrix}[/tex]
whose determinant is
[tex]\det\mathbf J=-\rho^2\sin\varphi[/tex]
Under this change of coordinates, we take
[tex]\mathrm dx\,\mathrm dy\,\mathrm dz=|\det\mathbf J|\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
So, we have
[tex]\displaystyle\iiint_{\mathcal W}\frac{\mathrm dx\,\mathrm dy\,\mathrm dz}{(x^2+y^2+z^2)^{3/2}}=\int_{\varphi=0}^{\varphi=\pi}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=r}^{\rho=R}\frac{\rho^2\sin\varphi}{(\rho^2)^{3/2}}\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
[tex]=\displaystyle2\pi\int_{\varphi=0}^{\varphi=\pi}\int_{\rho=r}^{\rho=R}\frac{\sin\varphi}\rho\,\mathrm d\rho\,\mathrm d\varphi[/tex]
[tex]=\displaystyle2\pi\left(\int_0^\pi\sin\varphi\,\mathrm d\varphi\right)\left(\int_r^R\frac{\mathrm d\rho}\rho\right)[/tex]
[tex]=2\pi\cdot2\cdot(\ln|R|-\ln|r|)[/tex]
[tex]=4\pi\ln\dfrac Rr[/tex]
