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MathGroup Archive 2003

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RE: Antiderivatives and Definite Integrals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg39713] RE: [mg39670] Antiderivatives and Definite Integrals
  • From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
  • Date: Mon, 3 Mar 2003 23:48:37 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

>-----Original Message-----
>From: Garry Helzer [mailto:gah at math.umd.edu]
To: mathgroup at smc.vnet.net
>Sent: Saturday, March 01, 2003 8:48 AM
>To: mathgroup at smc.vnet.net
>Subject: [mg39713] [mg39670] Antiderivatives and Definite Integrals
>
>
>The antiderivative of Sqrt[1+Cos[x]] discussed here recently (sorry, I 
>lost the thread) provides an amusing illustration of the fact that 
>Mathematica does not always evaluate definite integrals by first 
>finding an antiderivative and then substituting in the upper and lower 
>limits. (See the Mathematica book A.9.5) Make the definitions
>
>f[x_] = Integrate[Sqrt[1 + Cos[t]], {t, 0, x}]
>g[x_] := Integrate[Sqrt[1 + Cos[t]], {t, 0, x}]
>
>Then f[2Pi] is 0 (wrong) and g[2Pi] if 4Sqrt[2] (correct). Of course 
>f[x]==g[x] returns True.
>
>Garry Helzer
>Department of  Mathematics
>University of Maryland
>1303 Math Bldg
>College Park, MD 20742-4015
>
>

You might be interested to observe:

 
In[1]:= f[x_] = Integrate[Sqrt[1 + Cos[t]], {t, 0, x}]
Out[1]= 2*Sqrt[1 + Cos[x]]*Tan[x/2]

In[2]:= g[x_] := Integrate[Sqrt[1 + Cos[t]], {t, 0, x}]

In[3]:= ?g
Global`g
g[x_] := Integrate[Sqrt[1 + Cos[t]], {t, 0, x}]

In[4]:= Plot[f[x], {x, 0, 2Pi}] // Timing
Out[4]= {0.03 Second, = Graphics =}

...what you called "wrong".

In[5]:=
Plot[g[x], {x, 0, 2Pi}, PlotRange -> {All, {0, 6}}] // Timing
Out[5]= {4.096 Second, = Graphics =}

...what you called "right".
 

Redefine:

In[6]:= f[x_] /; -Pi < x < Pi = f[x]
Out[6]= 2*Sqrt[1 + Cos[x]]*Tan[x/2]

In[7]:= f[x_] /; Pi < x < 3 Pi = f[x] + 4*Sqrt[2]
Out[7]= 4*Sqrt[2] + 2*Sqrt[1 + Cos[x]]*Tan[x/2]


In[9]:=
Plot[f[x], {x, 0, 2Pi}, PlotRange -> {All, {0, 6}}] // Timing
Out[9]= {0.03 Second, = Graphics =}

Don't be confused by that double use of f (as a function, and as a shortcut
for an algebraic expression).

--
Hartmut Wolf



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