Re: Integrate vs. NIntegrate

• To: mathgroup at smc.vnet.net
• Subject: [mg51436] Re: Integrate vs. NIntegrate
• From: carlos at colorado.edu (Carlos Felippa)
• Date: Sun, 17 Oct 2004 03:05:37 -0400 (EDT)
• References: <ckqnbf\$nh0\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```carlos at colorado.edu (Carlos Felippa) wrote in message news:<ckqnbf\$nh0\$1 at smc.vnet.net>...
> Which one is the correct answer?
>
> f=2+Cos[2 x]+ I*Sin[2 x];
>
> Print[ N[Simplify[ Integrate[Log[f],{x,-Pi,Pi}]/(2*Pi)]] //InputForm];
> 0.34657359027997264 + 1.5707963267948966*I
>
> Print[            NIntegrate[Log[f],{x,-Pi,Pi}]/(2*Pi)   //InputForm];
> 0.6931471805599428 + 0.*I
>
> I believe the second one, have doubts about the first.

The 2nd answer  of Log[2] is verified by a brute force
Gauss-Legendre numerical integrator used in HWs:

NIntGauss3[fx_,x_,lower_,upper_,n_]:= Module[{fint,d,x1,x2,f,
g1=N[(1+Sqrt[3/5])/2],g2=N[1/2],g3=N[(1-Sqrt[3/5])/2]},
d=(upper-lower)/n; fint=0; f=fx;
Do[ x1=lower+(i-1)*d; x2=lower+i*d;
fint +=(N[5*f/.x->N[x1*g1+x2*g3]]+
N[8*f/.x->N[(x1+x2)/2]]+
N[5*f/.x->N[x1*g3+x2*g1]])*N[d/18],
{i,1,n}];
Return[fint]];

f=2+Cos[2 x]+ I*Sin[2 x];
Print[ NIntGauss3[Log[f],x,-Pi,Pi,200]/(2*Pi) //InputForm];
0.6931471805599455 - 3.969653887102225*^-17*I

Print[ NIntegrate[Log[f],{x,-Pi,Pi}]/(2*Pi)   //InputForm];
0.6931471805599428 + 0.*I

Print[ N[Log[2]] //InputForm];
0.6931471805599453

so NIntegrate[] gives 14 correct digits.

```

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