Re: Integral problem
- To: mathgroup at smc.vnet.net
- Subject: [mg27921] Re: Integral problem
- From: "Ian McInnes" <ian at whisper-wood.demon.co.uk>
- Date: Fri, 23 Mar 2001 04:31:57 -0500 (EST)
- References: <99cigc$8jq@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
The definitive check is to differentiate the result using D. This should
return the integrand (though in a possibly different form). Let:
mathint be the integral returned by Mathematica
otherint be the integral returned by the other system
(in Mathematica notation).
Then:
D[{mathint, otherint}, x]//Simplify
yields the integrand in each case, confirming that both results are correct.
An integrable function f[x] has is an infinite number of possible integrals,
due to an arbitrary integration constant (if some function F[x] is an
integral of f[x], then so is F[x] + c).
In fact, (mathint - otherint) is:
Pi/(4 Sqrt[3]) - Pi/12 I
This is a complex constant independent of the variable of integration.
However, I am not sure why Mathematica returns a much more complicated
expression than another system. The integral is a product of x^3 and (x^4 + x^2 +
1)^-1. The second factor is covered by standard tables, which give a result
similar to that from another system. Using the by parts formula with the other factor
x^3 yields a relatively simple result.
Also, Mathematica does not appear to be able to simplify the difference
between "mathint" and "otherint", although plotting the differences between
the real and imaginary components on a graph gives the above constant values
with only machine arithmetic errors.
Regards,
Ian McInnes.
"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
>