Solve Integration Homework: Prove Integral in Attachment

[zhulihuang]

Homework Statement

Please prove the integral in the attachment

Homework Equations

The Attempt at a Solution

Tried many ways to solve it. Integration by parts, trig sub etc.
The closest one is the substitution $U=\frac{r^2}{2}-\frac{R^2/2}+x^2$, which gives some arcsin. I change it into arctan, but it is not the arctan mathematica shows. I guess it has something to do with the $\sqrt{R^2 - x^2}$, since it will blow up at R. May need delta function to handle it.

 

[Simon Bridge]

Welcome to PF;

Please prove the integral in the attachment

Nice try :)
This is your homework - we can help you where you get stuck but you have to do the work.

Please give it a go and show your working as far as you get.
Try to describe how you are thinking about the problem as you go.
Don't worry about looking stupid - we've all done this before: you are among understanding, and friendly, people here.

 

[HallsofIvy]

You tried many ways- please show us at least some. If possible one where you feel you came closest to solving the problem. That will make it easier to give suggestions.

 

[Simon Bridge]

Tried many ways to solve it. Integration by parts, trig sub etc.
The closest one is the substitution $U=\frac{r^2}{2}-\frac{R^2}{2}+x^2$, which gives some $\arcsin$. I change it into $\arctan$, but it is not the $\arctan$ Mathematica shows. I guess it has something to do with the $\sqrt{R^2 - x^2}$, since it will blow up at $R$. May need delta function to handle it.

Well your integral (off the supplied attachment) seems to be: $$\int_0^b \frac{x.dx}{\sqrt{a^2+x^2}\sqrt{b^2-x^2}}=\frac{a\arctan(b/a)}{\sqrt{a^2}}$$ ... since there is no $r$ or $R$ in the integrand, it is difficult to see your reasoning behind the substitution. From context - I'm guessing R=b and r=a?

Considering the arctan in the solution - perhaps a substitution like $x=a\tan\theta$ and a table of trig identities will be the way to go?
Did you try that?

However, HallofIvy is right - it is better for you to go step-by-step through the one you felt got the closest.
So you made the substitution: what was the next step?

(I know: it's a pain typing out the math. Use the "quote" button off the bottom of this post and see how I was able to type the math out ... you can copy and paste then make minor adjustments.)