Re: Re: Re: Integral confusion
- To: mathgroup at smc.vnet.net
- Subject: [mg107345] Re: [mg107316] Re: [mg107297] Re: Integral confusion
- From: "David Park" <djmpark at comcast.net>
- Date: Wed, 10 Feb 2010 03:34:51 -0500 (EST)
- References: <hkm7d8$os0$1@smc.vnet.net> <201002080836.DAA29853@smc.vnet.net> <op.u7tlcde4tgfoz2@bobbys-imac.local> <22239832.1265704104854.JavaMail.root@n11>
Well, if you had Presentations you could use the Student's Integral section to manipulate the integral, for example break it into two integrals and factor out the minus sign, and then evaluate the resulting integrals from a basic integral table (instead of using the Mathematica Integrate command, although you can use that also.) You can also use basic integration techniques such as substitution of variable, complete the square, integration by parts and trigonometric substitution. This gives an experience and results more like that achieved in college calculus courses. Also, in this case you could also just map Integrate onto the two terms to obtain a more conventional result. David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: Nasser M. Abbasi [mailto:nma at 12000.org] The "weird" thing is not what specific constant of integration it returned, but the fact that it did return one at all. Integrate[x^2, x] gives x^3/2. not x^3/2+Pi*Sqrt[-1] or x^3/2+99 even though both are correct. > I don't know Integrate's steps in obtaining the result, but I'm sure it > didn't add extraneous constants deliberately. Ofcourse. I am sure of that too. But the final result is NOT what one would normally see in a textbook or expect or do when solving by hand, even though as I said, it is correct. It is just not "normal" looking. I am sorry, I did not mean to say anything bad about Mathematica when I said this result was "weird" even though it is correct. --Nasser ----- Original Message ----- From: "DrMajorBob" <btreat1 at austin.rr.com> To: "Nasser M. Abbasi" <nma at 12000.org>; <mathgroup at smc.vnet.net> Sent: Monday, February 08, 2010 9:54 AM Subject: [mg107345] [mg107316] Re: [mg107297] Re: Integral confusion >> But Mathematica does (normally?) return results for indefinite integrals >> without an explicit constant of integration. > > What's "explicit" or weird about the constants of integration in this > problem? > > I don't know Integrate's steps in obtaining the result, but I'm sure it > didn't add extraneous constants deliberately. > > Bobby > > On Mon, 08 Feb 2010 02:36:16 -0600, Nasser M. Abbasi <nma at 12000.org> > wrote: > >> >> "Jon Joseph" <josco.jon at gmail.com> wrote in message >> news:hkm7d8$os0$1 at smc.vnet.net... >>> All: Is this integral wrong? If not could someone explain the minus sign >>> inside the log? >>> >>> Integrate[1/(x + 1) - 1/(x + 6), x] // Simplify >>> >>> log(-2 (x + 1)) - log(2 (x + 6)) >>> >>> Thanks, Jon.= >>> >> >> Well, lets see: >> >> log(-2 (x + 1)) - log(2 (x + 6)) >> = log(-2)+log(1+x) -log(2)-log(x+6) >> = log(-1)+log(2)+log(1+x)-log(2)-log(x+6) >> = log(-1)+ log(1+x) - log(x+6) >> >> but log(-1) = Sqrt[-1]*Pi >> >> so result is >> >> Sqrt[-1]*Pi + log(1+x) - log(x+6) >> >> But we all know that the result should be >> >> log(1+x) - log(x+6) >> >> So, an extra term, Sqrt[-1]*Pi term pops up. But this term is a >> constant, so >> its derivative is zero, i.e. a constant of integration. >> >> Since >> >> D[log(-2 (x + 1)) - log(2 (x + 6)),x] will give back >> >> 1/(x + 1) - 1/(x + 6) >> >> So, in theory, the answer given by Mathematica is NOT wrong. >> >> But Mathematica does (normally?) return results for indefinite integrals >> without an explicit constant of integration. So I am not sure why it >> does in >> this case, and if it it does, why did not pick this constant? Why not >> C[1] >> as it does for DSolve[]? >> >> So, if I have to guess, I'd say this result is at least very weired, but >> mathematically it is not wrong? >> >> --Nasser >> >> >> > > > -- > DrMajorBob at yahoo.com
- References:
- Re: Integral confusion
- From: "Nasser M. Abbasi" <nma@12000.org>
- Re: Integral confusion