Re: Integrate vs NIntegrate

*To*: mathgroup at smc.vnet.net*Subject*: [mg89161] Re: [mg89129] Integrate vs NIntegrate*From*: danl at wolfram.com*Date*: Tue, 27 May 2008 07:16:39 -0400 (EDT)*References*: <200805261023.GAA15126@smc.vnet.net>

> Dear group, > > Why do I get different results (Res1, Res2) in Mathematica 6.0.1.0? > > *q=-1/2; a=0; b=3;* > > *h[x_]=(1+x^3)^q;* > > *f[x_]=Integrate[h[x],x];* > > *Res1=N[f[b]-f[a]]* > > *Res2=NIntegrate[h[x],{x,a,b}]* Why do you put in these asterisks? They make cut-and-paste a real pest. > ** > > *Res1 = -2.55387-2.42865 i* > > *Res2 = 1.65267* > > I'm assuming that Res2 is the correct answer. > > Regards, > > Armen The antiderivative, f[x], has a jump discontinuity due to a branch cut that is either at or very near to x=2 (probably at). Have a look at Plot[{Re[f[x]], Im[f[x]]}, {x, 0, 3}] Thus evaluation at the endpoints will fail to recover the definite integral. Integrate[h[x], {x, a, b}] will give a more plausible result, in accord with the NIntegrate value. Daniel Lichtblau Wolfram Research

**References**:**Integrate vs NIntegrate***From:*"Armen Kocharyan" <armen.kocharyan@gmail.com>