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MathGroup Archive 2007

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Re: Integration using mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72640] Re: Integration using mathematica
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Sat, 13 Jan 2007 04:30:02 -0500 (EST)
  • References: <eo7mt2$1ps$1@smc.vnet.net>

First things first!

In order to use Mathematica you should first learn the Basic Syntax of
it!

Here is a good place to start:
Copy/paste the following command in a notebook and execute it!

FrontEndExecute[{HelpBrowserLookup["MainBook", "T.0"]}]

You are right for example about the first letter being capitalized; but
on the other hand arguments of functions must be enclosed on square
brackets.

Apart from this your input misses some parentheses and I was not able
to figure out
exactly your expression!

Suppose I am right and I guess well, let f be the following expression

f = 0.013*Exp[15.16*ArcTan[0.002*x]^2]*ArcTan[0.002*x];

One thing you should learn for your very begining in using Mathematica
is not use inexact numbers for symbolic calculations like integration.
You can consult the archives of this forum if you want more details
about why.

A quick way to convert your real numbers to exact Rational numbers is
by using Rationalize.
Then

ff=Rationalize[f]
(13*E^((379/25)*ArcTan[x/500]^2)*ArcTan[x/500])/1000

Let's try to integrate (indefinite integral) this exression w.r.t. x.
We get

Integrate[ff, x]
(13*Integrate[E^((379/25)*ArcTan[x/500]^2)*ArcTan[x/500], x])/1000

Mathematica fails to return a analytical result.
There are three alternative explanations:

1) There is not a closed form expression. Especially for indefinite
integrals usually this is the
case within Mathematica.
2) Mathematica does not know the integral. Seldom especially for
indefinte integrals.
3) Mathematica "needs" a little help in order to determine the
integral.

Let know try definite integral of this expression. There are many
occasions that there is not a closed form expression for an indefinite
integral but there are closed form expressions for definite integrals.
For example

Integrate[BesselJ[0, x]/(1 + x^2), x]
Integrate[BesselJ[0, x]/(1 + x^2), x]

However,

Integrate[BesselJ[0, x]/(1 + x^2), {x, 0, Infinity}]
(1/2)*Pi*(BesselI[0, 1] - StruveL[0, 1])

Moreover, in definite integration you have also the numerical "way".

A quick check of the last result:

N[%]
0.8730842426508671

NIntegrate[BesselJ[0, x]/(1 + x^2), {x, 0, Infinity}, Method ->
Oscillatory]
0.8730842426508675

So, let integrate definite ff, say between 0 and Pi.
Before performing a definite integration (symbolic or numerical) always
plot the integrand for possible singularities in the intagrtion range.

Plot[(13*E^((379/25)*ArcTan[x/500]^2)*ArcTan[x/500])/1000, {x, 0, Pi},
PlotPoints -> 1000]

The graph, show normal behavior of the integrand. Warning, howver, that
the graph does
not guarranty that!

Anyway,

Integrate[ff, {x, 0, Pi}]
Integrate[(13*E^((379/25)*ArcTan[x/500]^2)*ArcTan[x/500])/1000, {x, 0,
Pi}]

Again no closed form result!

Finally,

NIntegrate[ff, {x, 0, Pi}]
0.00012834241443264268

I hope you will find my reply helfpful.
Try to repost your message with correct syntax if my guess was not
right!

Regards
Dimitris


Negede wrote:
> I have difficulty in integrating the following equation on Mathematica.
> Does any one of you know how to do it on Mathematica? Any trick to
> aply? or another software to use? Please help me. The function is as
> follows:
>
> 0.013 *Exp [15.16(ArcTan(0.002 x)])^2 *(ArcTan(0.002 x)
> 
> Thanks in advance


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