# Given the parameters of the electrostatics problem, is this integral possible to evaluate analytically? [closed]

A cone with apex at the origin has a height $$h$$ and a top radius $$h$$, a uniform charge density with no charge on the top face.

I need to find the potential $$V$$ at a position $$z$$ on the cone's axis using spherical coordinates. I'm told I won't be able to evaluate the $$r'$$ integral analytically so my answer will have the integral expression.

I've done the steps in the photo (please see) but I don't see why I can't evaluate the integral at the end (could use wolframalpha.com for example). Please assist me - what did I do wrong? Or is my work correct? For the final integral, first complete the square in the denominator to get $${r'^2+z^2-\sqrt 2 zr'}=(r'-\frac{\sqrt2}{2}z)^2+\frac{z^2}{2}$$. The integral then is: $$\int_0^h dr' \frac{r'}{\sqrt{(r'-\frac{\sqrt2}{2}z)^2+\frac{z^2}{2}}}$$ Now use the substitution $$r' - \frac{z}{\sqrt2} = \frac{z}{\sqrt 2}\tan u$$ to get: $$\int_{-\pi/4}^{\tan^{-1}(\sqrt 2h/z - 1)}\frac{\frac {z^2} {2}(\tan u + 1)(1+\tan^2u)}{\frac{z}{\sqrt 2}(1+\tan^2u)^{1/2}}du$$ $$=\frac{z}{\sqrt 2}\int_{-\pi/4}^{\tan^{-1}(\sqrt 2h/z - 1)}{(\tan u + 1)(1+\tan^2u)^{1/2}}du$$ $$=\frac{z}{\sqrt 2}\int_{-\pi/4}^{\tan^{-1}(\sqrt 2h/z - 1)}{\frac{(\tan u + 1)}{|\cos u|}}du$$ $$=\frac{z}{\sqrt 2}\int_{-\pi/4}^{\tan^{-1}(\sqrt 2h/z - 1)}{\frac{(\tan u + 1)}{\cos u}}du$$ $$=\frac{z}{\sqrt 2}\int_{-\pi/4}^{\tan^{-1}(\sqrt 2h/z - 1)}{\frac{\sin u}{\cos^2 u}}du+\frac{z}{\sqrt 2}\int_{-\pi/4}^{\tan^{-1}(\sqrt 2h/z - 1)}{\sec{u} \ }du$$ The first integral is easily solved by the substitution $$x=\cos u$$, turning it into a $$\int \frac{dx}{x}=\ln x +c$$ type integral. The second integral is a simple secant integral $$\int\sec u du = \ln|\sec u + \tan u|+c$$.
I've left the final calculations for you to go through. Also note that you've written $$z = r \cos \theta = r \cos 0 = r$$ on the top of the page which is incorrect ($$\theta = \pi/4$$). I would just write the potential as a function of $$z$$, so you just have to replace $$r$$ with $$z$$ in your calculations (which is what I've done above), since you've incorrectly assumed that $$r=z$$.