Need Help with Elementary integral derivation.

[FriendlyHippo]

Hello! :)

Having a hard time managing to solve ad integral in Landau Theoretical Physics Volume 1 page 28 (3rd edition) (Determination of the potential energy from the period of oscillation)

The integral is ∫1/√(E-u)(σ-E) with respect to dE between u and sigma. The book claims that the integral is elementary and should come to equal ∏, yet no matter how many different ways I have attempted this, I have drawn a blank. Any help on the correct derivation would be much appreciated.

 

[FriendlyHippo]

Heres an example of this integral on wolfram-alpha. No explanation is given, and I just can't find a way of doing it.

http://www.wolframalpha.com/input/?i=integrate+1/(sqrt(x-y)+*+sqrt(z-x))+dx+between+y+and+z

 

[Ray Vickson]

Heres an example of this integral on wolfram-alpha. No explanation is given, and I just can't find a way of doing it.

http://www.wolframalpha.com/input/?i=integrate+1/(sqrt(x-y)+*+sqrt(z-x))+dx+between+y+and+z

If 0 < a < b, let $$J(a,b) =\int_a^b \frac{1}{\sqrt{(x-a)(b-x)}}\, dx.$$ First change variables to y = x-a, so the integral runs from y = 0 to y = b' = b-a; now a is replaced by 0 and b is replaced by b'. Finally, let z = y/b'. Write out the z-integral and see what you get.

RGV

 

[FriendlyHippo]

now a is replaced by 0 and b is replaced by b'.

Allowing for this, the integral come out as expected, thanks. I have not come across this method for the quote above before. Could you possibly point me in the right direction as to why it is allowed. As setting something within the integral to zero seems like it could be a useful technique for other times, that I would like understand.

Thanks

Hippo