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MathGroup Archive 2007

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message question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74682] message question
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Sat, 31 Mar 2007 01:30:51 -0500 (EST)

Clear["Global`*"]

$Version
"5.2 for Microsoft Windows (June 20, 2005)"

The settings

Integrate[f[x], x]

and

Derivative[-1][f[#1] & ][x]

give (one of the possible)  antiderivative of f[x].

E.g.

Integrate[Cos[x], x]
Sin[x]

Derivative[-1][Cos[#1] & ][x]
Sin[x]

Other examples

foolst = {BesselJ[n, x], Log[Cos[x]]*Sin[x], 1/Sqrt[1 + x^2]};

Integrate[foolst, x]
FullSimplify[D[%, x]]

{2^(-1 - n)*x^(1 + n)*Gamma[(1 + n)/
2]*HypergeometricPFQRegularized[{(1 + n)/2}, {1 + n, (3 + n)/2}, -
(x^2/4)],
  Cos[x] - Cos[x]*Log[Cos[x]], ArcSinh[x]}
{BesselJ[n, x], Log[Cos[x]]*Sin[x], 1/Sqrt[1 + x^2]}

foolst2 = foolst /. x -> #1;

Derivative[-1][foolst2 & ][x]
{2^(-1 - n)*x^(1 + n)*Gamma[(1 + n)/
2]*HypergeometricPFQRegularized[{(1 + n)/2}, {1 + n, (3 + n)/2}, -
(x^2/4)],
  Cos[x] - Cos[x]*Log[Cos[x]], ArcSinh[x]}

To my surprise for the following function we get a warning message
even though the result
is correct!

Integrate[Log[Sin[x]^2]*Tan[x], x]
Simplify[D[%, x]]

Log[Sec[x/2]^2]^2 + 2*Log[Sec[x/2]^2]*Log[(1/2)*Cos[x]*Sec[x/2]^2] +
  Log[Sec[x/2]^2]*Log[Sin[x]^2] - 2*Log[Sec[x/2]^2]*Log[-1 + Tan[x/
2]^2] -
  Log[Sin[x]^2]*Log[-1 + Tan[x/2]^2] + Log[Tan[x/2]^2]*Log[-1 + Tan[x/
2]^2] +
  2*PolyLog[2, (1/2)*Sec[x/2]^2] + PolyLog[2, Cos[x]*Sec[x/2]^2] +
  PolyLog[2, -Tan[x/2]^2]
Log[Sin[x]^2]*Tan[x]


Derivative[-1][Log[Sin[#1]^2]*Tan[#1] & ][x]
D[%,x]//Simplify

(*the messages*)
General::patop: Pattern Integrate`TableDump`b$_. /; ! \
Internal`DependsOnQ[{Integrate`TableDump`b$, Integrate`TableDump`n$},
1] \
contains inappropriate optional object.
General::patop: Pattern Integrate`TableDump`b$_. /; ! \
Internal`DependsOnQ[{Integrate`TableDump`b$, Integrate`TableDump`n$},
1] \
contains inappropriate optional object.

(*correct result*)
Log[Sec[x/2]^2]^2 + 2*Log[Sec[x/2]^2]*Log[(1/2)*Cos[x]*Sec[x/2]^2] +
Log[Sec[x/2]^2]*Log[Sin[x]^2] -
  2*Log[Sec[x/2]^2]*Log[-1 + Tan[x/2]^2] - Log[Sin[x]^2]*Log[-1 +
Tan[x/2]^2] + Log[Tan[x/2]^2]*Log[-1 + Tan[x/2]^2] +
  2*PolyLog[2, (1/2)*Sec[x/2]^2] + PolyLog[2, Cos[x]*Sec[x/2]^2] +
PolyLog[2, -Tan[x/2]^2]
Log[Sin[x]^2]*Tan[x]


Does anyone understand the presence of the warning messages?

I see the message documentation but I can't understand its appearance
here!

Thanks a lot!
Dimitris



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