Re: Bug of Integrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg82817] Re: Bug of Integrate*From*: m.r at inbox.ru*Date*: Wed, 31 Oct 2007 06:22:28 -0500 (EST)*References*: <fg4dfv$6c3$1@smc.vnet.net><fg6pse$d44$1@smc.vnet.net>

On Oct 30, 2:26 am, "David W.Cantrell" <DWCantr... at sigmaxi.net> wrote: > Miguel <misv... at gmail.com> wrote: > > When I try to calculate the integral > > > Integrate[Sqrt[1/Cos[t]^2]*3*Cos[t],{t,0,2Pi}] Mathematica 6.0.1 > > yields -6*Pi. > > If so, then that is clearly a bug. But in version 5.2, the result is > correct: > > In[1]:= Integrate[Sqrt[1/Cos[t]^2]*3*Cos[t],{t,0,2Pi}] > > Out[1]= 0 > > > Simplifying the expresion resultrs Integrate[3,{t,0,2*Pi}] and It is > > clear that the correct solution is 6*Pi. > > No. Simplifying the integrand does not give 3, rather > > In[2]:= Simplify[Sqrt[1/Cos[t]^2]*3*Cos[t], Element[t,Reals]] > > Out[2]= 3 Abs[Sec[t]] Cos[t] > > which BTW is the same as 3 Sign[Cos[t]] for real t except when Cos[t]==0. > > However, related to the above, version 5.2 does give an incorrect result > for a definite integral with a symbolic real limit. Whether this error > still exists in version 6, I don't know: > > In[3]:= Assuming[Element[x,Reals],Integrate[3*Sign[Cos[t]],{t,0,x}]] > > Out[3]= 3 If[x > 0, x Abs[Cos[x]] Sec[x], > Integrate[Sign[Cos[t]], {t, 0, x}, Assumptions -> x <= 0]] > > The above is incorrect for x > Pi/2. A correct result would have been > > 3 Sign[Cos[x]] (x - Pi Floor[x/Pi + 1/2]) > > for all real x. > > David W. Cantrell Note that your formula isn't correct for x = Pi/2 + Pi k. The correct expression for all real x is In[1]:= Assuming[0 <= x < 2 Pi, Integrate[3 Sign[Cos[t]], {t, 0, x}]] /. x -> Mod[x, 2 Pi] Out[1]= Piecewise[{{-3 Pi/2, Mod[x, 2 Pi] == 3 Pi/2}, {3 (Pi - Mod[x, 2 Pi]), Pi/2 < Mod[x, 2 Pi] < 3 Pi/2}, {-3 (2 Pi - Mod[x, 2 Pi]), 3 Pi/ 2 < Mod[x, 2 Pi] < 2 Pi}, {3 Mod[x, 2 Pi], 0 < Mod[x, 2 Pi] <= Pi/2}}] Maxim Rytin m.r at inbox.ru