Infinite definite integral

[Thomas Klemm]

(11-08-2014 11:40 PM)lrdheat Wrote:  What prompts this is int(x/(1-sqrt(x))) from 0 to 1.

\(\int\frac{x}{1-\sqrt{x}}dx=-\frac{1}{3}\sqrt{x}(2x+3\sqrt{x}+6)-2\log(1-\sqrt{x})+C\)

Quote:I'm amazed at how close to 1 one can get, and still have a fairly small area, far from infinity....

That's due to the last term: \(-2\log(1-\sqrt{x})\)
For \(x=1-\epsilon\) we get:
\(-2\log(1-\sqrt{1-\epsilon})\approx-2\log(1-(1-\frac{1}{2}\epsilon))=-2\log(\frac{1}{2}\epsilon)\)

This grows slowly as \(\epsilon \to 0\).

Quote:from 0 to 1-1EE12 produces something 50 or 60ish!

With \(\epsilon=10^{-12}\) we get approximately:
\(-\frac{11}{3}+2(\log(2)+12\log(10))\approx52.982\)

Kind regards
Thomas