Re: Integrate question

*To*: mathgroup at smc.vnet.net*Subject*: [mg82308] Re: Integrate question*From*: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>*Date*: Wed, 17 Oct 2007 04:05:15 -0400 (EDT)*References*: <ff1pru$924$1@smc.vnet.net> <4714D9C2.5090506@gmail.com>

Jean-Marc Gulliet wrote: > Oskar Itzinger wrote: > > > Mathematica 5.2 under IRIX complains that > > > > Integrate[x/(3 x^2 - 1)^3,{x,0,1}] > > > > doesn't converge on [0,1]. > > > > However, Mathematica 2.1 under Windows gives the corrrect answer, (1/16). > > > > When did Mathematica lose the ability to do said integral? > > FWIW, Mathematica for Windows 5.2 as well as 6.0.1 cannot do it either, > although it does not seem that hard to get the correct answer (the > indefinite integral and the limits are evaluated in a breeze). > > In[1]:= int = Integrate[x/(3 x^2 - 1)^3, x] > > Out[1]= -(1/(12 (-1 + 3 x^2)^2)) > > In[2]:= Limit[int, x -> 1] - int /. x -> 0 > > Out[2]= 1/16 Sorry to have posted this nonsense (I was too quick to answer without thinking). Indeed, Mathematica 2.1 is wrong (as I was in my previous reply) in claiming that the definite integral is equal to 1/16, for the integrand has a non removable discontinuity at 1/Sqrt[3] (about 0.55) and each integral are divergent on the interval {0,1/Sqrt[3]} (-Infinity) and {1/Sqrt[3],1} (Infinity). In[1]:= Plot[x/(3 x^2 - 1)^3, {x, 0, 1}] Solve[12 (-1 + 3 x^2)^2 == 0, x] % // N Integrate[x/(3 x^2 - 1)^3, {x, 0, 1/Sqrt[3]}] Integrate[x/(3 x^2 - 1)^3, {x, 1/Sqrt[3], 1}] Out[1]= [ ... graphic deleted ... ] Out[2]= {{x -> -(1/Sqrt[3])}, {x -> -(1/Sqrt[3])}, {x -> 1/Sqrt[ 3]}, {x -> 1/Sqrt[3]}} Out[3]= {{x -> -0.57735}, {x -> -0.57735}, {x -> 0.57735}, {x -> 0.57735}} During evaluation of In[1]:= Integrate::idiv: Integral of x/(-1+3 \ x^2)^3 does not converge on {0,1/Sqrt[3]}. >> Out[4]= Integrate[x/(-1 + 3*x^2)^3, {x, 0, 1/Sqrt[3]}] During evaluation of In[1]:= Integrate::idiv: Integral of x/(-1+3 \ x^2)^3 does not converge on {1/Sqrt[3],1}. >> Out[5]= Integrate[x/(-1 + 3*x^2)^3, {x, 1/Sqrt[3], 1}] Sorry for the confusion, -- Jean-Marc