Re: Bug of Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg82835] Re: Bug of Integrate
- From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
- Date: Thu, 1 Nov 2007 05:08:26 -0500 (EST)
- References: <fg4dfv$6c3$1@smc.vnet.net> <fg6pse$d44$1@smc.vnet.net> <fg9pmb$n1a$1@smc.vnet.net>
m.r at inbox.ru wrote: > 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. Yes. Many thanks for pointing that out! I should have been more careful. > 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}}] That is _one_ correct expression for all real x. But there are shorter ones, such as the one which I should have posted originally: 3 ArcSin[Sin[x]] David W. Cantrell