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Re: Please explain this weird Integrate result

• To: mathgroup at smc.vnet.net
• Subject: [mg60149] Re: Please explain this weird Integrate result
• From: "Carl K. Woll" <carlw at u.washington.edu>
• Date: Sun, 4 Sep 2005 03:02:16 -0400 (EDT)
• Organization: University of Washington
• References: <dfbevv$inu$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

"Jose Reckoner" wrote:
>
> Expand[Integrate[(2*x)/(2*x - 1), x]]
>
> gives
>
> -1/2 + x + Log[-1 + 2*x]/2
>
> But,
>
> Expand[Integrate[1 + 1/(-1 + 2*x), x]]
>
> gives
>
> x + Log[-1 + 2*x]/2
>
> even though
>
> (2*x)/(2*x - 1) == 1 + 1/(-1 + 2*x)
>
> Where did the extra -1/2 in the first result come from?
>

Indefinite integrals are only defined up to a constant, hence there is 
nothing wrong mathematically going on here. Check that the integrals are 
correct by taking derivatives:

In[11]:=
D[-1/2 + x + Log[-1 + 2*x]/2,x]//Together
Out[11]=
(2 x)/(-1+2 x)

In[12]:=
D[x + Log[-1 + 2*x]/2,x]//Together
Out[12]=
(2 x)/(-1+2 x)

Carl Woll
Wolfram Research