Integrate vs NIntegrate

• To: mathgroup at smc.vnet.net
• Subject: [mg46868] Integrate vs NIntegrate
• From: "Mukhtar Bekkali" <mbekkali at hotmail.com>
• Date: Fri, 12 Mar 2004 02:02:53 -0500 (EST)
• Organization: Iowa State University
• Sender: owner-wri-mathgroup at wolfram.com

```I am confused why NIntegrate misbehaves on such a simple function as mine.

Here is what I have:

In:

f=D[1/(1+(1+(a-b)^2)),a];
g=Integrate[f*b*(1-b)^2,{b,0,1},Assumptions->0<a<1];
FindRoot[g==0,{a,0,1}]

Out:

a->0.397207

However, since Integrate takes long, I tried to use NIntegrate instead and
this is what I get

In:

f=D[1/(1+(1+(a-b)^2)),a];
g:=NIntegrate[f*b*(1-b)^2,{b,0,1}];
FindRoot[g==0,{a,0,1}]

Out:

a->1

or, FindRoot+NIntegrate give me the upper boundary of a.  If I abandon the
secant method and turn to Newton, i.e. use
FindRoot[g==0,{a,0.5}] instead then I get the message that Jacobian is
singular at a=0.5 and get no solution.  Perturbing the starting value of a
does not help.

What is going on here?

PS.  Is there a way to get M5 to tackle the problem where:
(1) I define some function f[x]:=NIntegrate[g[x,y],{y,0,1}], then
(2) take the derivative of f[x] with respect to x, say h[x]:=f'[x] and then
(3) Use FindRoot to find x such that h[x]==0

```

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