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improper integral error?

asked 2015-03-11 08:33:19 -0500

rafarob gravatar image

Is there an alternative way to calculate this improper integral?

integral(log(x)/(x^2 - 1),x,0,infinity)

the integral is convergent to pi^2/4 but Sage says divergent.

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answered 2015-03-11 09:16:02 -0500

Always try the other available "algorithms":

sage: integral(log(x)/(x^2 - 1),x,0,infinity, algorithm='mathematica_free')
-1/2*log(x + 1)*log(x) - 1/2*polylog(2, -x) - 1/2*polylog(2, -x + 1)

See http://sagemath.org/doc/reference/cal...

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Though this algorithm requires the internet. Interestingly, using algorithm='sympy' gives AttributeError: 'NonElementaryIntegral' object has no attribute '_sage_'.

kcrisman gravatar imagekcrisman ( 2015-03-11 10:06:55 -0500 )edit

Unfortunately, the mathematica_free algorithm will not work from within Sage Math Cloud with current connection settings.

calc314 gravatar imagecalc314 ( 2015-03-11 10:10:25 -0500 )edit

But the correct answer is a numerical value or exact expression pi^2/4. So, algorithm = 'mathematica_free' doesn't work.

rafarob gravatar imagerafarob ( 2015-03-11 11:39:28 -0500 )edit
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answered 2015-03-12 09:44:13 -0500

calc314 gravatar image

With the algorithm set to mathematica_free, Sage is returning the antiderivative even though you are asking for the improper integral. You can get the improper integral by doing the following:

f=integral(log(x)/(x^2 - 1),x,algorithm='mathematica_free')
limit(f,x=oo)-limit(f,x=0)

This returns pi^2/4.

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Asked: 2015-03-11 08:33:19 -0500

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Last updated: Mar 11 '15