I have an integral
NIntegrate[1/(x - x*Log[x]), {x, 0, E}, Method -> "PrincipalValue",
Exclusions -> {0, E}, PrecisionGoal -> 10, WorkingPrecision -> 13];
and I can't solve it. Can anyone take a look and show me how to do it?
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$\begingroup$ I get $58.605...$. Do you have any reason to doubt that answer? $\endgroup$– David G. StorkCommented Mar 28, 2018 at 18:14
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$\begingroup$ @David G. Stork: Yes, I do. The answer in version 11.3 is supplied with NIntegrate obtained \ 61.6898965566571980569209393874979487231066384596339583582607001 and \ 1.87176935662578407544374941801080191407846253122765831122636127 for \ the integral and error estimates $\endgroup$– user64494Commented Mar 28, 2018 at 18:27
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$\begingroup$ @user64494 MMA 11.2 returns an answer $61.68...$, together with warnings about slow convergence. Can you explain exactly what your problem is though? $\endgroup$– MarcoBCommented Mar 28, 2018 at 18:34
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$\begingroup$ @MarcoB: Could you kindly look at my comment to the Sungmin's answer? $\endgroup$– user64494Commented Mar 28, 2018 at 18:41
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1$\begingroup$ I'm voting to close this question as off-topic because the issue it raises is not really a Mathematica issue but a matter of the OP not having grasped the mathematics involved. $\endgroup$– m_goldbergCommented Apr 3, 2018 at 12:30
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2 Answers
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Your integral can be done symbolically:
Integrate[1/(x-x Log[x]), {x, 0, E}, PrincipalValue->True]
∞
answered Mar 28, 2018 at 19:26
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$\begingroup$ Integrate[1/(x - x Log[x]), {x, 0, E}] says Integrate::idiv: Integral of 1/(x-x Log[x]) does not converge on {0,E}. $\endgroup$ Commented Mar 28, 2018 at 20:29
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This is more or less a comment. It seems that the integral diverges.
f[x_] = Integrate[ 1/(x - x*Log[x]), x]
r = Assuming[ {\[Epsilon] > 0},
Series[ f[E - \[Epsilon]] - f[\[Epsilon]], {\[Epsilon], 0, 1}]
]
r // Normal // ReplaceAll[\[Epsilon] -> 0]
(* \[Infinity] *)
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$\begingroup$ Yes, the singularity at $x=0$ causes the divergence in view of Series[1/(x - x*Log[x]), {x, 0, 2}] which outputs $$\frac{1}{x (1-\log (x))}+O\left(x^4\right). $$ $\endgroup$ Commented Mar 28, 2018 at 18:40
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$\begingroup$ Can you tell me why some integrals, for instance NIntegral of x^-0.5 from 0 to 1 is converging. Although there is singularity in 0? $\endgroup$ Commented Mar 28, 2018 at 19:46