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Re: Re: Bug? Analytical integration of cosines gets the sign wrong

• To: mathgroup at smc.vnet.net
• Subject: [mg107372] Re: [mg107346] Re: Bug? Analytical integration of cosines gets the sign wrong
• From: Louis Talman <talmanl at gmail.com>
• Date: Thu, 11 Feb 2010 05:17:21 -0500 (EST)
• References: <hkeb9k$b5$1@smc.vnet.net> <201002100835.DAA21213@smc.vnet.net>

Actually, something very strange---even stranger than has already been  
pointed out---is going on.

Try, in version 7 of Mathematica (be prepared to wait a little while):

--------------------
f[x_] := Integrate[Cos[ t ] Cos[ 2 t ] Cos[ 4 t ], { t, Pi, x }]

Plot[ f[x], {x, Pi, 2 Pi}, PlotRange -> All ]
--------------------



Then compare the resulting plot with the one obtained from

--------------------
F[x_] = Integrate[ Cos[ x ] Cos[ 2 x ] Cos[ 4 x ], x]

Plot[ F[x] - F[Pi], { x, Pi, 2 Pi}, PlotRange -> All ]
--------------------

On my PPC iMac running OS X, v10.5.8, the first plot is the reflection  
of the second plot about the x-axis.


On Feb 10, 2010, at 1:35 AM, WetBlanket wrote:

> On Feb 4, 5:33 am, K <kgs... at googlemail.com> wrote:
>> Hello everyone,
>>
>> the analytical integration in Mathematica 7.01.0 on Linux x86 (64bit)
>>
>> faultyInt =
>> Integrate[Cos[ph]*1/Pi*Cos[4*ph]*Cos[2*ph], {ph, Pi, 3/2*Pi}]
>>
>> gives as result:
>>
>> 19/(105 \[Pi])
>>
>> which is as a decimal number
>>
>> N[faultyInt]
>>
>> 0.0575989
>>
>> The numerical integration
>>
>> NIntegrate[Cos[ph]*1/Pi*Cos[4*ph]*Cos[2*ph],{ph,Pi,3/2*Pi}]
>>
>> gives
>>
>> -0.0575989
>>
>> which I believe is correct by judging from the plot
>>
>> Plot[Cos[ph]*1/Pi*Cos[4*ph]*Cos[2*ph], {ph, Pi, 3/2*Pi},
>> PlotRange -> {-1/Pi, 1/Pi}]
>>
>> and because the quadgk function in another system gives the same
>> negative result.  Could anyone try this at home (or work, rather)
>> and confirm or disprove it?
>> Thanks,
>> K.
> If One substitutes the sequence, {1.9, 1.99, 1.999, ... 2.0}
> Mathematica gets the correct answer for as long as I tried except, of
> course for 2.0.
>

• References:
  • Re: Bug? Analytical integration of cosines gets the sign wrong
    • From: WetBlanket <wyvern864@gmail.com>