How to calculate $\int_{0}^1 \sqrt{\frac{1-x^2}{1+x^2}}dx$?

Asked Aug 13 '16 at 07:19

Active Aug 13 '16 at 08:33

Viewed 126 times

I encounter the following question: $$\int_{0}^1 \sqrt{\frac{1-x^2}{1+x^2}}dx=?$$ It looks simple. But I doubt there exists analytic solution. Could anyone help? Thank you!

Icarus 369 has pointed out Page [ Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$ ]. The question indeed has no elementary solution.

Thanks Icarus 369.

The elliptic integration turns out to be very interesting. FYI: http://web.mst.edu/~lmhall/SPFNS/sfch3.pdf

edited Apr 13 '17 at 12:20

Community

asked Aug 13 '16 at 07:19

kaiwu

@BehrouzMaleki, have tried, but get a mess of formulae, which can not be calculated. – kaiwu Aug 13 '16 at 07:23
Have you tried the change of variable : $x=sin(\theta)$ ? – Jean Marie Aug 13 '16 at 07:24
@JeanMarie $dx=\cos(\theta) d\theta$ becomes a trouble term – kaiwu Aug 13 '16 at 07:28
The indefinite integral can't be solved using elementary functions.(the solution involves eliptic integral). – Icarus 369 Aug 13 '16 at 07:31
Numerical Integration $\simeq0.7$ – Behrouz Maleki Aug 13 '16 at 07:31
The change of variable I indicated has the merit to make the radical disappear... a classical objective in most algebraic computations when it is possible. – Jean Marie Aug 13 '16 at 07:34
@BehrouzMaleki@lcarus 369, Thanks. I try to check whether 0.7.. is some combination of pi and rationals to get some hints. But sadly, failed .... – kaiwu Aug 13 '16 at 07:35
@JeanMarie ??.... – kaiwu Aug 13 '16 at 07:38
I beg your pardon. The radical doesn't disappear. – Jean Marie Aug 13 '16 at 07:44
1

http://math.stackexchange.com/questions/1410439/evaluating-int-sqrt-frac1-x21x2-mathrm-dx?noredirect=1&lq=1 – Icarus 369 Aug 13 '16 at 07:48
@Icarus 369, Thanks! – kaiwu Aug 13 '16 at 07:59
This has no closed form. Wolfram gives the answer as $2K(-1)-E(-1)$ where $K(n)$ is the Complete Elliptic Integral of the First Kind and $E(n)$ is the Elliptic Integral of the Second Kind. – Ian Miller Aug 13 '16 at 08:09
you can probably use $u=sin(2x)$ also. – Our Aug 13 '16 at 08:50

How to calculate $\int_{0}^1 \sqrt{\frac{1-x^2}{1+x^2}}dx$?

0 Answers0