Re: Integral problem
- To: mathgroup at smc.vnet.net
- Subject: [mg27923] Re: [mg27874] Integral problem
- From: Tomas Garza <tgarza01 at prodigy.net.mx>
- Date: Fri, 23 Mar 2001 04:32:02 -0500 (EST)
- References: <200103220930.EAA08509@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
The answers only seem to be totally different. The problem is that no
previous information is given as to the nature of x. If x is real, there is
no problem, and your integrand behaves nicely. However, if x is allowed to
be complex, then you have 4 complex zeroes of the denominator which have to
be taken care of. So, Mathematica gives
In[1]:=
a1 = Integrate[x^3/(x^4 + x^2 + 1), x] // Simplify
Out[1]=
1/(4*(3*I + Sqrt[3]))*((2 - 2*I*Sqrt[3])*
ArcTan[(-1 + x^2)/(Sqrt[3]*(1 + x^2))] +
4*ArcTan[(2 + x^2)/(Sqrt[3]*x^2)] +
(3*I + Sqrt[3])*Log[1 + x^2 + x^4])
and the other "symbolic algebra systems" give
In[2]:=
a2 = 1/4 Log[x^4 + x^2 + 1] - (Sqrt[3]/6)* ArcTan[(2x^2 + 1)/Sqrt[3]]
But, then
In[3]:=
D[a1, x] // Simplify
Out[3]=
x^3/(1 + x^2 + x^4)
and, of course,
In[4]:=
D[a2, x] // Simplify
Out[4]=
x^3/(1 + x^2 + x^4)
so that the two derivatives coincide, as one would expect. What, then? The
problem, as pointed out above, arises when x is not real. Take the real part
of a1,
and you'll see that, as long as x is real, it is equal to
In[5]:=
a3 = Simplify[Map[Re, a1, 6], x \[Element] Reals]
Out[5]=
(4*ArcTan[(1 + 2/x^2)/Sqrt[3]] +
2*ArcTan[(-1 + x^2)/(Sqrt[3]*(1 + x^2))] +
Sqrt[3]*Log[1 + x^2 + x^4])/(4*Sqrt[3])
which differs from a2 by a constant term Pi/(2*Sqrt[3]) (the constant of
integration). To check on this, compare
In[6]:=
Series[-(Pi/(2*Sqrt[3])) + Sqrt[3]/6*
ArcTan[(2*x^2 + 1)/Sqrt[3]], {x, 0, 20}]
with
In[7]:=
1/(4*Sqrt[3])*Series[4*ArcTan[(1 + 2/x^2)/Sqrt[3]] +
2*ArcTan[(-1 + x^2)/(Sqrt[3]*(1 + x^2))], {x, 0, 20}]
(I omit the output) where you can see that the two series are identical.
Tomas Garza
Mexico City
----- Original Message -----
From: "Jose Lasso" <jml at accessinter.net>
To: mathgroup at smc.vnet.net
Subject: [mg27923] [mg27874] Integral problem
> Hello,
>
> Well in my calculus class, I need to integrate the following expression:
> (x^3/(x^4+x^2+1))dx, I solve the integral with Mathematica, but a few
> classmates got a different answer using other symbolic algebra
> system, the answers are totally different, the answer that my
> classmates got is:
> 1/4 Ln(x^4+x^2+1)-(Sqrt(3)/6) ArcTg((2x^2+1)/Sqrt(3)) is this the
> correct answer?? Thx in advance. Regards
>
> Jose M Lasso
>
- References:
- Integral problem
- From: Jose Lasso <jml@accessinter.net>
- Integral problem