Symbolic integration  Printable Version + HP Forums (https://archived.hpcalc.org/museumforum) + Forum: HP Museum Forums (https://archived.hpcalc.org/museumforum/forum1.html) + Forum: Old HP Forum Archives (https://archived.hpcalc.org/museumforum/forum2.html) + Thread: Symbolic integration (/thread151459.html) 
Symbolic integration  Valentin Albillo  06092009 Hi all, Quite an interesting subject, most especially for small handhelds with symbolic integration capabilities. For those who don't know about it, this online symbolic integrator might be of interest to try some cases and check the results and capabilities against those of your favourite handheld: 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 16character in length.
Edited: 9 June 2009, 9:53 a.m.
Re: Symbolic integration  Jorge I. Rodriguez (Pasadena, CA)  06092009 Very nice integrals. I will check them out in both the HP50g and the HP35s and see what happens.
Re: Symbolic integration  Quan  06092009 Quote: If I understand correctly, the 35s doesn't do symbolic integration.
Am I wrong?
Re: Symbolic integration  Egan Ford  06092009 Your 50g will run out of memory. It does work on another handheld. The iPhone.
Re: Symbolic integration  Jorge I. Rodriguez (Pasadena, CA)  06092009 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 HP50g gives or what Mathematica gives.
Re: Symbolic integration  Karl Schneider  06102009 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 closedform 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 n1 terms in the solution.  KS
Edited: 10 June 2009, 3:36 a.m.
Re: Symbolic integration  Marcus von Cube, Germany  06102009 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.
