[Date Index]
[Thread Index]
[Author Index]
Re: Please explain this weird Integrate result
*To*: mathgroup at smc.vnet.net
*Subject*: [mg60146] Re: Please explain this weird Integrate result
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Sun, 4 Sep 2005 03:02:11 -0400 (EDT)
*Organization*: The Open University, Milton Keynes, U.K.
*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?
>
Hi Jose,
the constant term -1/2 is a constant of integration. Say f(x) is any
real function, F(x) is the primitive of f(x), then any function F(x) +
C, C being an arbitrary constant, is an indefinite integral of f(x);
that is (F(x) + C)' = f(x).
Now, *Integrate* does not return any symbolic constant term such as C
(so we would get x + (1/2)*Log[-1 + 2*x] + C for instance), nor does it
return, usually, any numeric constant term (or you may think of the
value of the constant to be zero).
However, a constant term might be returned sometimes, like in your case.
Two different, although equivalent, expressions may lead to different
representations of the solution since the algorithms used by *Integrate*
might be different in each case.
As long as you work with indefinite integrals, you may "discard" the
constant term -1/2 and be sure that both indefinite integrals represent
the same family of functions.
In[1]:=
int1 = Expand[Integrate[(2*x)/(2*x - 1), x]]
Out[1]=
-(1/2) + x + (1/2)*Log[-1 + 2*x]
In[2]:=
int2 = Expand[Integrate[1 + 1/(-1 + 2*x), x]]
Out[2]=
x + (1/2)*Log[-1 + 2*x]
In[3]:=
int1 - int2
Out[3]=
-(1/2)
In[4]:=
D[int1, x]
Out[4]=
1 + 1/(-1 + 2*x)
In[5]:=
D[int2, x]
Out[5]=
1 + 1/(-1 + 2*x)
Obviously, the second form for the integrand is the one preferred by
Mathematica.
Best regards,
/J.M.
Prev by Date:
**Re: my wish list for Mathematica next major version**
Next by Date:
**Re: Re: piecewise vs which**
Previous by thread:
**Re: Please explain this weird Integrate result**
Next by thread:
**ComplexExpand confusion**
| |