# Possible error in solving Improper integral **URL:** \ **Category:** Questions\ **Tags:** calculus, numerical-computation\ **Created:** [April 2, 2025, 8:48am UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329 "2025-04-02T08:48:31Z")\ **Posts on this page:** 11\ **Page:** 1
**Author:** ![juniotor2023](https://avatars.discourse-cdn.com/v4/letter/j/a5b964/32.png) [@juniotor2023](https://community.wolfram.com/u/juniotor2023)\ **Post date:** [April 2, 2025, 8:48am UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/1 "2025-04-02T08:48:31Z")
NIntegrate[ArcTan/(x^2 - x + 1), {x, 0, Infinity}] ``` 1.89941 Integrate[ArcTan[x]/(x^2 - x + 1), {x, 0, Infinity}] // N ``` 1.89941 - 3.64593 I !!! ``` Integrate[ArcTan[x]/( x^2 - x + 1), {x, 0, Infinity}] -> ( x -> (1 - y)/(1 + y )) -> Integrate[(2 ArcTan[(1 - y)/(1 + y)])/(1 + 3 y^2), {y, -1, 1}] -> 2 Integrate[(Pi/4 - ArcTan[y])/(1 + 3 y^2), {y, -1, 1}] -> Pi/2 Integrate[1/(1 + 3 y^2), {y, -1, 1}] - 2 Integrate[ArcTan[y]/(1 + 3 y^2), {y, -1, 1}] -> \[Pi]^2/( 3 Sqrt[3]) - 0 N[\[Pi]^2/(3 Sqrt[3])] ``` 1.89941 [improper-integral.nb](https://community.wolfram.com/uploads/short-url/x1ThudcNqGVHnAUI1mUiZhKrRXl.nb) (6.97 KB) ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [April 2, 2025, 1:08pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/2 "2025-04-02T13:08:01Z")
Let us try a workaround. The primitive given by Mathematica is discontinuous: ``` f[x_] = Integrate[ArcTan[x]/(x^2 - x + 1), x] Plot[f[x], {x, 0, Infinity}] ``` This is where the discontinuity occurs: ``` FunctionDiscontinuities[f[x], x, Complexes]; FullSimplify[%, Element[x, Reals]] ``` The full integral can be calculated this way: ``` disc = 2 + Sqrt[3]; f0 = f[0]; fdisc0 = Limit[f[x], x -> disc, Direction -> "FromBelow"]; fdisc1 = Limit[f[x], x -> disc, Direction -> "FromAbove"]; finfinity = Limit[f[x], x -> Infinity]; finfinity - fdisc1 + fdisc0 - f0 // FullSimplify % // N ``` ---
**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\ **Post date:** [April 2, 2025, 5:31pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/3 "2025-04-02T17:31:45Z")
This appears to be a bug in `Integrate`. ---
**Author:** ![mariuszi28cg](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/mariuszi28cg/32/308_2.png) [@mariuszi28cg](https://community.wolfram.com/u/mariuszi28cg)\ **Post date:** [April 2, 2025, 6:31pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/4 "2025-04-02T18:31:39Z")
This comment can be deleted. ---
**Author:** ![mariuszi28cg](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/mariuszi28cg/32/308_2.png) [@mariuszi28cg](https://community.wolfram.com/u/mariuszi28cg)\ **Post date:** [April 2, 2025, 6:52pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/5 "2025-04-02T18:52:05Z")
Another way: [Open in Wolfram Cloud ↗](https://www.wolframcloud.com/obj/b3dad58b-8eb4-4ce9-b5a9-c2d12e72caa5) ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [April 3, 2025, 10:08am UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/6 "2025-04-03T10:08:25Z")
It seems that `FullSimplify` changes the numerical value of an expression a lot: ``` In[69]:= a = finfinity - fdisc1 + fdisc0 - f0; a // N FullSimplify[a] // N Out[70]= -0.474852 - 1.19435 I Out[71]= 1.89941 + 0. I ``` ---
**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\ **Post date:** [April 3, 2025, 3:48pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/7 "2025-04-03T15:48:01Z")
`FullSimplify` might not have changed the numeric value at all. What might instead have changed is their numerical conditioning. This can be checked by using e.g. `N[...,50]`. ---
**Author:** ![juniotor2023](https://avatars.discourse-cdn.com/v4/letter/j/a5b964/32.png) [@juniotor2023](https://community.wolfram.com/u/juniotor2023)\ **Post date:** [April 4, 2025, 4:45am UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/8 "2025-04-04T04:45:28Z")
In[21]:= Integrate[ArcTan/(x^2 - x + 1), {x, 0, Infinity}] // FullSimplify ``` Out[21]= 1/( 288 Sqrt[ 3]) (121 \[Pi]^2 - 36 (ArcCos[2]^2 - 4 Log[2] Log[28 - 16 Sqrt[3]] + 2 Log[8 - 4 Sqrt[3]]^2 + 4 (PolyLog[2, 1 - (-1)^(1/6)] + PolyLog[2, 1/2 ((2 + I) - Sqrt[3])] + PolyLog[2, 2/((2 - I) + Sqrt[3])] + PolyLog[2, 2/((2 + I) + Sqrt[3])]))) In[22]:= N[%] Out[22]= 1.89941 - 3.20494*10^-17 I ``` ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [April 4, 2025, 11:14am UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/9 "2025-04-04T11:14:35Z")
Is it `Accuracy` that does a poor job here? ``` int = Integrate[ArcTan[x]/(x^2 - x + 1), {x, 0, Infinity}]; int // N % // Accuracy ``` ---
**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\ **Post date:** [April 4, 2025, 2:10pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/10 "2025-04-04T14:10:05Z")
When you used `N` without a second argument, it works with machine numbers and `Accuracy/Precision` are not really meaningful since they are not tracked. ---
**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\ **Post date:** [April 4, 2025, 2:18pm UTC](https://community.wolfram.com/t/possible-error-in-solving-improper-integral/25329/11 "2025-04-04T14:18:06Z")
Jose, What you show using `FullSimplify` is fine of course. I believe there may be a version- or timing-dependent bug wherein it can give an inequivalent result. Yours is correct and the use of simplification also makes it numerically “nice” (as in well-conditioned). I may need to file a report about this, if I find that it is incorrect in our development version. — edit — I have filed a report. — end edit —