I was doing a physics problem which leads me to this equation:

[tex]

V = \frac{\sigma}{4\pi \epsilon_0}\int_{0}^{R}\int_{0}^{2\pi} \frac{r}{\sqrt{R^2 + r^2 - 2Rr\cos{\theta}}} \,d\theta \,dr

[/tex]

Stumped by the math, I turned to the solution. However, without a word of explanation, the solution jumps from the above equation to the following:

[tex]

V = \frac{\sigma}{4\pi \epsilon_0}R\int_{0}^{2\pi} \cos{\theta} \ln{\left ( 1 + \frac{\sqrt{2}}{\sqrt{1-\cos{\alpha}} \right ) }\,dA +8R - 2\pi R

[/tex]

I had no idea how that happened. I would appreciate it if someone could explain what I should to do arrive at this latter equation. I don't need a full explanation, just a push in the right direction.

EDIT: I understand that it's an integral, I just don't understand how the integral was done. Specifically, this one:

[tex]

\int_{0}^{R} \frac{r}{\sqrt{R^2 + r^2 - 2Rr\cos{\theta}}} \,dr

[/tex]

Thanks.

[tex]

V = \frac{\sigma}{4\pi \epsilon_0}\int_{0}^{R}\int_{0}^{2\pi} \frac{r}{\sqrt{R^2 + r^2 - 2Rr\cos{\theta}}} \,d\theta \,dr

[/tex]

Stumped by the math, I turned to the solution. However, without a word of explanation, the solution jumps from the above equation to the following:

[tex]

V = \frac{\sigma}{4\pi \epsilon_0}R\int_{0}^{2\pi} \cos{\theta} \ln{\left ( 1 + \frac{\sqrt{2}}{\sqrt{1-\cos{\alpha}} \right ) }\,dA +8R - 2\pi R

[/tex]

I had no idea how that happened. I would appreciate it if someone could explain what I should to do arrive at this latter equation. I don't need a full explanation, just a push in the right direction.

EDIT: I understand that it's an integral, I just don't understand how the integral was done. Specifically, this one:

[tex]

\int_{0}^{R} \frac{r}{\sqrt{R^2 + r^2 - 2Rr\cos{\theta}}} \,dr

[/tex]

Thanks.

Last edited: