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 &#8805; 0 && L &#8805; 0 && n &#8805; 0]
\!\(If[C > 0 && L > 0, L\^n\ \@\(C\ L\)\ \@&#960;\  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 &#8804; 0 || L &#8804; 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,
Attributes[C] = {NHoldAll, Protected}

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).

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