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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






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