Re: Integral problem
- To: mathgroup at smc.vnet.net
- Subject: [mg27931] Re: Integral problem
- From: "Allan Hayes" <hay at haystack.demon.co.uk>
- Date: Sat, 24 Mar 2001 00:48:56 -0500 (EST)
- References: <99cigc$8jq@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Several postings have already pointed out that both solutions provide an
antiderivative:
expr=x^3/(x^4+x^2+1);
Mma=Integrate[expr,x];
D[Mma,x]-expr//Simplify
0
Oth= 1/4 Log[x^4+x^2+1]-(Sqrt[3]/6) ArcTan[(2x^2+1)/Sqrt[3]];
D[Oth,x]-expr//Simplify
0
The following plots may be of interest: I generated them to see how the
complex plain is carved up by the two solutions.
DensityPlot[Evaluate[Im[Oth /. x -> u + I*v]],
{u, -1.5, 1.5}, {v, -1.5, 1.5}, PlotPoints -> 250,
Mesh -> False]
DensityPlot[Evaluate[Re[Oth /. x -> u + I*v]],
{u, -1.5, 1.5}, {v, -1.5, 1.5}, PlotPoints -> 250,
Mesh -> False]
DensityPlot[Evaluate[Im[Mma /. x -> u + I*v]],
{u, -1.5, 1.5}, {v, -1.5, 1.5}, PlotPoints -> 250,
Mesh -> False]
DensityPlot[Evaluate[Re[Mma /. x -> u + I*v]],
{u, -1.5, 1.5}, {v, -1.5, 1.5}, PlotPoints -> 250,
Mesh -> False]
( the singularities of expr are given by
Solve[Denominator[expr]\[Equal]0,x]
{{x -> -(-1)^(1/3)}, {x -> (-1)^(1/3)},
{x -> -(-1)^(2/3)}, {x -> (-1)^(2/3)}}
N[%]
{{x -> -0.5000000000000001 - 0.8660254037844386*I},
{x -> 0.5000000000000001 + 0.8660254037844386*I},
{x -> 0.4999999999999998 - 0.8660254037844388*I},
{x -> -0.4999999999999998 + 0.8660254037844388*I}}
)
--
Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565
"Jose Lasso" <jml at accessinter.net> wrote in message
news:99cigc$8jq at smc.vnet.net...
> 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
>