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 Discussion: All Activities in Calculus on Computer Algebra System Topic: Intigration

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 Subject: RE: Intigration Author: Si Date: May 5 2006
On May  5 2006, Alan Cooper wrote:

> The example we started with *is* a good one, because the integrals
> that arise in applied problems (especially but not exclusively eg in
> time value of money problems, in electric discharge problems, and in
> the use of Laplace transforms to solve differential equations) often
> involve combinations of exponential and algebraic functions for
> which there is no closed form answer in terms of elementary
> functions.

I appologise if my question has been answered by someone in the past, but is
this forum mainly for high school related mathematic discussion
? I am not a teacher, perhaps I am a wannabe teacher (the appropriate one), my
only experience in teaching is coaching Maths for high school students, mainly
children of Family members and friends (and their friends). Over the past years,
I have taken external level math exam students who were hopeless (parents
believed that they were destined to fail no matter what) and coached them into a
level of understanding they passed. Some, I use software tools to help, and some
would not use software at all. I recognised that the ones who I used software,
are those that if left alone, they would find the answer themselves, so I
believe that those students are very confident and love maths. The others who
I've tried to use software but not very keen, are the ones I noticed that are
very low confident in themselves. In my view, they just like to be guided by a
tutor.

Now back to symbolic algebra. The integration problem presented is a University
level math problem, and that is why I asked at the beginning if this forum is
for highschool level maths or not. If it is  mainly highschool, then the student
needs to be taught to recognise certain things. One of those first simple rule
for them to take notice of such integration problem is to ask themselves : 1)
Can there be a closed form solution ?  2) If the answer is NO, then there is no
other  way but to attack it via Numerical Methods. Runge-Kutta methods is one
of the popular ones. But again, Runge-Kutta (RK) is a university level topic
and I would be surprised if high school are teaching such complex topic. However
RK is available in most symbolic & numeric math tools of today.

e^(-8*x) /(x^3*(4*x+1))

The next step for a student is to ask. Does the quotient integration function
contain any poles (discontinuity). I noticed that the problem given contain
poles at x=0 (repeated 3 times) and at x=-1/4. With the numerator term as an
exponential ( e^(-8*x) ), the problems is very much harder for a highschool
level math student to comprehend.

Cheers,
Sione.