So I was trying to calculate this equation $\int \sin\varphi\sqrt{r^2-2rz\cos\varphi+z^2}\,d\varphi$, which I plug into calculator and get the result as $\frac{(r^2-2rz\cos\varphi+z^2)^{\frac{3}{2}}}{3rz}$. But having division by $rz$ seems wrong to me, cause for example let $z=0$, the original equation collapses into $\int (\sin\varphi) r\,d\varphi$, which is clearly integrate-able, while the original result will encounter division by 0.
And if looking from geometrical interpretation of that integral, it'll be the sum of distance from all points that are distance $r$ away from origin to the point located at $(0,0,z)$ (this coordinate is cartesian coordinate). So by setting $z=0, r>0$, it should equal to the volume of a sphere with radius=r.
And if you look at the graph of the function we're integrating:
The area under that curve is definitely not infinity, so why the computer returns that result? And is that result correct? If so, then how could we calculate values where $z=0$ (cause the result doesn't converges to a value when $z$ approaches 0).