Infinite definite integral
11-09-2014, 05:22 AM
Post: #4
 Thomas Klemm Senior Member Posts: 1,886 Joined: Dec 2013
RE: Infinite definite integral
(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
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 Messages In This Thread Infinite definite integral - lrdheat - 11-08-2014, 11:40 PM RE: Infinite definite integral - lrdheat - 11-08-2014, 11:51 PM RE: Infinite definite integral - Paul Dale - 11-09-2014, 12:03 AM RE: Infinite definite integral - Thomas Klemm - 11-09-2014 05:22 AM

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