Symbolic integration



#8

Hi all,

Quite an interesting subject, most especially for small handhelds with symbolic integration capabilities.

For those who don't know about it, this on-line symbolic integrator might be of interest to try some cases and check the results and capabilities against those of your favourite handheld:

Wolfram Online Integrator

I suggest you try this example I've concocted, to see why many times straight numerical approximations are far more preferable than using an exact symbolic expression, even if it exists in terms of elementary functions:


    1/(1+x^1024)

You won't get a "Traditional Form" so try either the "Input Form" or "Output Form" and be astonished at the result. How does your handheld fare with my example ?

If it doesn't fare, try reducing the exponent from 1024 to 512, 256, 128, 64, ... , until it can cope.

Another nice example to try, this time resulting in no trigonometrics at all, is:


    1/(1+x^(1/4096))

Again, marvel at the result's length and complexity, most especially when compared to those of the input, a mere 16-character in length.


Best regards from V.

Edited: 9 June 2009, 9:53 a.m.


#9

Very nice integrals. I will check them out in both the HP-50g and the HP-35s and see what happens.


#10

Quote:
Very nice integrals. I will check them out in both the HP-50g and the HP-35s and see what happens.

If I understand correctly, the 35s doesn't do symbolic integration.

Am I wrong?

#11

Your 50g will run out of memory. It does work on another handheld. The iPhone.


#12

You are right the HP35s does not do symbolic integration, but I can try putting limits on the integrand and do numerical integration to see if the answer matches what HP-50g gives or what Mathematica gives.

#13

Welcome back, Valentin!

For integrands of the form

1/(1+x^n)

those who remember their calculus should be able to easily solve the integral for n = 1 or 2. For n = 3 or n = 4, looking up the closed-form solutions of cubic and quartic equations, substituting the roots into the denominator, expressing the integrand as a sum of partial fractions, and performing the elementary integrations should provide the answer. However, this gets tedious right away.

For the symbolic solutions for n > 4, I suppose that Mathematica is doing something like the following procedure found at the following link:

http://mathforum.org/library/drmath/view/53784.html

With n as an integer power of 2, there might possibly be simplifications, but maybe not. The sin, cos, sec, and csc terms are present for every solution with n >= 7, as far as I looked.

The number of terms in the integral generally equals the order of the polynomial, as might be expected from the basic approach described above. Exception: When one term is a scalar multiple of tan-1(x), as with n = 2, 6, 10, ...(?), there are only n-1 terms in the solution.

-- KS


Edited: 10 June 2009, 3:36 a.m.

#14

Hi Valentin,

I could not resist and put it into Derive 6. At least, Derive is capable of solving the integral. On my iMac (with Windows XP in a Parallels virtual machine), processing time was 276 seconds!

I'll probably check with the TI Nspire later.

Edit: I'll probably leave the Nspire alone. ;) I'm just trying with the Nspire software on my iMac which is busily working on the n=16 case for a few minutes now without a result so far. The same case is solved by Derive almost instantaneously, even while the Nspire software is eating resources.

Edited: 10 June 2009, 2:50 a.m.


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