       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:= Integrate[Sec[x], x]

Out= 2*ArcTanh[Tan[x/2]]

In:= (% /. x -> 4*Pi/3) - (% /. x -> 2*Pi/3)

Out= -4*ArcTanh[Sqrt]

In:= FullSimplify[ComplexExpand[%]]

Out= 2*(I*Pi + Log[-1 + Sqrt] - Log[1 + Sqrt])

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 is not continuous on that interval.
In other words, Out 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:= Integrate[Sec[x], {x, 2*Pi/3, 4*Pi/3}]

Out= -2*I*Pi - 4*ArcTanh[Sqrt]

In:= FullSimplify[ComplexExpand[%]]

Out= 2*(Log[-1 + Sqrt] - Log[1 + Sqrt])

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:= Integrate[Sec[x], {x, 0, Pi}, PrincipalValue -> True]

Out= 0

In:= Integrate[Sec[x], {x, Pi, 2*Pi}, PrincipalValue -> True]

Out= 0

In:= Integrate[Sec[x], {x, 0, 2*Pi}, PrincipalValue -> True]

Out= 2*I*Pi

So this is an example of an incorrect definite integral of Sec[x].

David W. Cantrell

```

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