Re: Re: Solving difficult integral a follow up

*To*: mathgroup at smc.vnet.net*Subject*: [mg18747] Re: [mg18665] Re: [mg18620] Solving difficult integral a follow up*From*: Richard Gass <gass at physics.uc.edu>*Date*: Sat, 17 Jul 1999 02:36:56 -0400*Sender*: owner-wri-mathgroup at wolfram.com

>> Date: Thu, 15 Jul 1999 01:45:51 -0400 >> From: Richard Gass <gass at physics.uc.edu> To: mathgroup at smc.vnet.net >> To: mathgroup at smc.vnet.net >> Subject: [mg18747] [mg18665] Re: [mg18620] Solving difficult integral a follow up >> >> Christian wrote >> <I've got the following problem: >> >> <I need to solve the following integral: >> >> <Integrate[Sqrt[C0+C1*x+C2*x^2+C3*x^3+C4*x^4],{x,0,t}] >> >> <For those of you who do not recognize it at once: it's the integral of the >> <square root of a 4th degree polynomial in x. I've tried to solve the above >> <integral with Mathematica, but after, say, two hours of computing I stopped >> <the calculation - it looked like the Kernel crashed or whatever. Is >>there an >> <analytical solution to this problem at all? >> >> I replied that I could do the indefinite integral but that I thought the >> definite integral was hopeless because of branch cut problems. However >> my machine ( a fairly old Power Mac) run 4.0 returns an answer for >> Integrate[Sqrt[C0 + C1*x + C2*x^2 + C3*x^3 + C4*x^4], {x, 0, t}]. THe >> result takes a few minutes or so. I am not at all sure the result is >> correct. >> Richard Gass >> Department of Physics >> University of Cincinnati >> Cincinnati, OH 45221 >> phone- 513-556-0519 >> E-Mail gass at physics.uc.edu >> > >My Mac (a G3 that's about a year old) will do the symbolic integral, too. It >results in about 13 pages of Root Objects intermixed with Elliptic >functions and >Arcsins. However, when I substitute the value 1 for each of the coefficients >and try to plot the resulting integral over the interval from 0 to 5, >Mathematica v4 responds with "not a machine-size real number" messages and >produces no plot. This in spite of the fact that 1 + x + x^2 + x^3 + x^4 >is a >positive real number everywhere. > >--Lou Talman > Department of Mathematical and Computer Sciences > Campus Box 38 > Metropolitan State College of Denver > PO Box 173362 > Denver CO 80217-3362 > > http://clem.mscd.edu/~talmanl I get the same result that you do when I try this. If I look at the expression I find that it is full of terms like (-1)^(1/5), (-1)^(2/5) and so on. I think that they arise because the "Wrong" root has been picked by the integration routines . If for example we try Solve[x^5==1] we get five solutions only one of which is real. I think the integration routines have picked up one of the complex roots instead of the real root. Richard Gass Department of Physics University of Cincinnati Cincinnati, OH 45221 phone- 513-556-0519 E-Mail gass at physics.uc.edu