Integration in the spherical coordinate

\[ d^3r = r^2 dr \sin\theta d\theta d\phi \]

Since $d\cos(\theta)=-\sin(\theta)d\theta$,

\[ d^3r = -r^2 dr d\cos(\theta) d\phi ,\]

let $\mu = \cos(\theta)$,

\[ \int \int \int_0^{\pi} d^3r =  \int \int \int_0^{\pi} -r^2 dr d\cos(\theta) d\phi \\  =  \int \int \int_{\cos(0)}^{\cos(2\pi)} -r^2 dr d\mu d\phi \\ =\int \int \int_{\cos(0)}^{\cos(2\pi)} -r^2 dr d\mu d\phi \\ = \int \int \int_{\cos(2\pi)}^{\cos(0)} r^2 dr d\mu d\phi \\       = \int \int \int_{-1}^{1} r^2 dr d\mu d\phi \]