Re: Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !

*To*: mathgroup at smc.vnet.net*Subject*: [mg70237] Re: Closed-form Integral Solution without Hypergeometric2F1Regularized ! ! !*From*: dimmechan at yahoo.com*Date*: Sun, 8 Oct 2006 02:05:04 -0400 (EDT)*References*: <eg845g$nfd$1@smc.vnet.net>

Let me state that your post is quite unreadable. Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, Assumptions -> C ≥ 0 && L ≥ 0 && n ≥ 0] \!\(If[C > 0 && L > 0, L\^n\ \@\(C\ L\)\ \@π\ Gamma[1 + n]\ Hypergeometric2F1Regularized[\(-\(1\/2\)\), 1 + n, 3\/2 + n, \(-\(L\/C\)\)], Integrate[x\^n\ \@\(\(C + x\)\/\(L - \x\)\), {x, 0, L}, Assumptions -> C ≤ 0 || L ≤ 0]]\) Are you sure that the input and output match together? I have my reasons to object this. I got Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, (Assumptions -> C & )*#8805; (0 && L & )*#8805; (0 && n & )*#8805; 0] Integrate::"ilim" : "Invalid integration variable or limit(s) in \!\(0\) ." Integrate[x^n*Sqrt[(C + x)/(L - x)], {x, 0, L}, 0] Anyway, I will try to say a few things. First of all avoid to use capital letters if you are on introductory level with Mathematica. There might be great problems because of possible conflicts with built-in symbols. Above you use C. However C is a built in symbol. Information["C", LongForm -> True] "C[i] is the default form for the ith parameter or constant generated in representing the results of various symbolic computations."*Button[More..., ButtonData :> "C", Active -> True, ButtonStyle -> "RefGuideLink"] Attributes[C] = {NHoldAll, Protected} Here is your integral Integrate[x^n*Sqrt[(c + x)/(l - x)], {x, 0, l}] Mathematica 5.2 fails to evaluate the integral (or to state better, I with 5.2 fail to evaluate the integral; I believe it must be a way to get the desired result). In Mathematica 4.0 I took the following result Integrate[x^n*Sqrt[(c + x)/(l - x)], {x, 0, l}] (l^(1 + n)*Sqrt[(c + l)/l]*Sqrt[Pi]*Gamma[2 + n]* Hypergeometric2F1[1 + n, -(1/2), 3/2 + n, -(l/c)])/ (Sqrt[(c + l)/c]*(1 + n)*Gamma[3/2 + n]) which look a little strange on me since I wait a conditional result... By the way I want to know what version you use. Supposing now that the last result of 4.0 is correct (you must check it numerically; always check Integrate with NIntegrate) you can get even further FullSimplify[%] (l^(1 + n)*Sqrt[(c + l)/l]*Sqrt[Pi]*Gamma[1 + n]*Hypergeometric2F1Regularized[1 + n, -(1/2), 3/2 + n, -(l/c)])/Sqrt[(c + l)/c] On this stage, you must understand that Hypergeometric2F1Regularized is indeed a CLOSED FORM solution proper and approachable to every kind of calculation (e.g. plotting). I will be very glad to help you more but this time send your post in more readable format. It is quite easy. After you finish with your inputs/outputs select the cells and press Shift+Ctrl+I and you will get everything in Input format. Then Copy/Paste on your message. Regards Dimitris