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Re: Trouble with an integral


On 5/26/05 at 4:31 AM, astanoff at yahoo.fr (Valeri Astanoff) wrote:

>Could someone give me the right answer for this integral with
>parameter 'a' ?

>In[1]:=$Version

>Out[1]=5.0 for Microsoft Windows (November 18, 2003)

>In[2]:=Integrate[Sqrt[1+Sqrt[x]]/(2*Sqrt[2]*E^(a*x)*Sqrt[x]),{x,0,1
>},Assumptions -> a>0]

>Out[2]=(1/3)*(4 - Sqrt[2])

>In[3]:=%//N

>Out[3]=0.861929

>In[4]:=NIntegrate[Sqrt[1+Sqrt[x]]/(2*Sqrt[2]*E^(2*x)*Sqrt[x]),{x,0,
>1}]

>Out[4]=0.491465

By inspection, it is clear the function you are integrating is positive over the range 0<=x<=1 and has greater values for the case where a = 1 as compared to when a = 2. That means the value of the integral cannot be independent of the value of a, making the result returned by Integrate incorrect.

Given

In[10]:=NIntegrate[Sqrt[1 + Sqrt[x]]/(2*Sqrt[2]*E^(0*x)*Sqrt[x]), {x, 0, 1}]
Out[10]=
0.8619288125480162

It appears Integrate evaluated the integral with a = 0 rather than for a > 0.
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