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