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MathGroup Archive 2004

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Re: Nasty bug in Integrate (version 5.0)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg45914] Re: [mg45881] Nasty bug in Integrate (version 5.0)
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 29 Jan 2004 05:36:04 -0500 (EST)
  • References: <200401281019.FAA18530@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Mathematica seems to get lost in the large number of checks that it 
attempts to perform to find the exact conditions  (in the complex 
plane) for the convergence of this integral. Even if a and b are 
assumed to be real there is a problem that 0 may lie in between them. 
Only in the case when you specify that the limits are both positive or 
both negative you get the correct answers because Mathematica can 
quickly deal with this problem.
If you really want to see how much checking Mathematica attempts to do 
just evaluate:

Trace[Integrate[1/x + x^c, {x, a, b}], TraceInternal->True]

and be prepared to wait and obtain a huge output.

Anyway, this is clearly a  bug. However, it is in general a good idea 
not to ask a computer program to do what is trivial to do by hand, 
which in this case means evaluating a definite rather than an 
indefinite integral. Note that


Integrate[1/x + x^c, x]

x^(c + 1)/(c + 1) + Log[x]

and the answer is obtained instantly. If this is what you really were 
after (you can substitute in the limits for example by using:


Subtract @@ (Integrate[1/x + x^c, x] /. x -> #1 & ) /@ {b, a}


-(a^(c + 1)/(c + 1)) - Log[a] + Log[b] + b^(c + 1)/(c + 1)


Of course this now makes sense only under certain conditions on a,b and 
c which you now have to determine yourself.


On 28 Jan 2004, at 11:19, Math User wrote:

> Hi,
>
> Sorry if this has been discussed before. Could anyone explain why
> Mathematica 5.0 is ignoring the term 1/x in the first and second
> answers?
>
> Thanks!
>
>
> In[1]:= Integrate[1/x + x^c, {x, a, b}]
>
> Out[1]= (-a^(1 + c) + b^(1 + c))/(1 + c)
>
> In[2]:= Integrate[1/x + x^c, {x, a, b}, Assumptions -> {Element[a,
> Reals], Element[b, Reals]}]
>
> Out[2]= (-a^(1 + c) + b^(1 + c))/(1 + c)
>
> In[3]:= Integrate[1/x + x^c, {x, a, b}, Assumptions -> {a > 0, b > 0}]
>
> Out[3]= (-a^(1 + c) + b^(1 + c))/(1 + c) + Log[b/a]
>
>


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