Antiderivative to the formula f(x)=(2xr-x²)^(1/2)

The anti-derivative you are seeking is:

[tex]
\frac{2 r^2 \sqrt{2 r-x} \sqrt{x} \tan ^{-1}\left(\frac{\sqrt{x}}{\sqrt{2 r-x}}\right)-x \left(2 r^2-3 x r+x^2\right)}{2 \sqrt{(2
r-x) x}}
[/tex]

I highly recommend this site:

http://integrals.wolfram.com/index.jsp

And I highly recommend the software package Mathematica that is made by this company.

 

Ouch. That integral hurts a lot. I wonder how to even begin doing it.

 

well off hand it just looks like sqrt(a^2 - u^2), after completing the square, which would make it an arcsin.