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Re: Fundamental theorem problem
*To*: mathgroup at smc.vnet.net
*Subject*: [mg49613] Re: Fundamental theorem problem
*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>
*Date*: Sat, 24 Jul 2004 03:48:26 -0400 (EDT)
*References*: <cdqqnb$km6$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
"Steven Jonak" <jonakst at gw.kirkwood.k12.mo.us> wrote:
> I input the command: D[Integrate[Sec[t],{t,1,x^4}],x] expecting to get
> 4x^3 Sec[x^4] but instead got a fairly complicated result that doesn't
> resemble what one would expect from the Fundamental Theorem of Calculus.
> What am I doing wrong? Help!
In another response, I noted that there appears to be no problem if an
appropriate assumption (x^4 < Pi/2) is used. But I now realize that there
are troubles with integrating the secant function. Consider the following:
In[1]:= Integrate[Sec[x], x]
Out[1]= 2*ArcTanh[Tan[x/2]]
In[2]:= (% /. x -> 4*Pi/3) - (% /. x -> 2*Pi/3)
Out[2]= -4*ArcTanh[Sqrt[3]]
In[3]:= FullSimplify[ComplexExpand[%]]
Out[3]= 2*(I*Pi + Log[-1 + Sqrt[3]] - Log[1 + Sqrt[3]])
What's wrong? This result (based on the fundamental theorem) should be
real, but it is not. How did this happen? Well, despite the fact that
Sec[x] is continuous between 2*Pi/3 and 4*Pi/3, and thus has a continuous
antiderivative on that interval, Out[1] is not continuous on that interval.
In other words, Out[1] is not an antiderivative on that interval, although
such an antiderivative does exist. Mathematica has _unnecessarily_
introduced a spurious discontinuity.
Thankfully however, Mathematica does the definite integral below correctly.
In[4]:= Integrate[Sec[x], {x, 2*Pi/3, 4*Pi/3}]
Out[4]= -2*I*Pi - 4*ArcTanh[Sqrt[3]]
In[5]:= FullSimplify[ComplexExpand[%]]
Out[5]= 2*(Log[-1 + Sqrt[3]] - Log[1 + Sqrt[3]])
Let's also look at some Cauchy principal values. The first two are
correctly 0. The third should be the sum of the first two, and so 0 also,
but it is not.
In[6]:= Integrate[Sec[x], {x, 0, Pi}, PrincipalValue -> True]
Out[6]= 0
In[7]:= Integrate[Sec[x], {x, Pi, 2*Pi}, PrincipalValue -> True]
Out[7]= 0
In[8]:= Integrate[Sec[x], {x, 0, 2*Pi}, PrincipalValue -> True]
Out[8]= 2*I*Pi
So this is an example of an incorrect definite integral of Sec[x].
David W. Cantrell
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