Symbolic integration on HP48GX



#5

Even if I tried it seems that my 48 GX isn't able to solve the very simple symbolic itegration of the function sin(x)*cos(x). The same function is solved indeed by a very primitive and buggy program like Derive. Does anyone know something more about it??

Thanks Aldo


#6

I know, that I'm in danger making myself completely ridiculous with answering in such a thread, but anyway, I just can't resist.

First the solution: (I'll write the integral sign as an upper-case I here.)

I sin(x) * cos(x) * dx

with: I u * dv = u * v - I v * du i'll write

u = sin(x) => du = cos(x) * dx dv = cos(x) * dx => v = sin(x)

I sin(x) * cos(x) * dx = sin(x) * sin(x) - I cos(x) * sin(x) *dx

rearranging the right Integral gives:

I sin(x) * cos(x) * dx = sin(x) * sin(x) - I sin(x) * cos(x) *dx

I add I sin(x) * cos(x) *dx to both sides of the equation.

2 * (I sin(x) * cos(x) * dx ) = sin(x) * sin(x)

which leads to

I sin(x) * cos(x) * dx = ( sin(x) * sin(x) ) / 2

I hope this formats well on the forum.

I don't have my 48GX manual handy, but the 48 SX manual says, thet the calculator compares PATTERNS of math strings for integration. This makes clear, that it does integration of polynomials (a rather easy task) or simple (even nested) functions, but no substitutions like the one above.

I'm pretty sure, that Derive has a better engine for integrating symbolic expressions than the 48.

The 48SX manual even gives an example of an expression, that can't be integrated, but can be formed, so that integration is possible after some operations. (I have the German version, your's might differ, my fellow Americans)

I fear, there's no way around learning such things the hard way (with lot's of integrals solved by hand) and even then, there's not always the right (or right formulated) result.

Just take it easy. Did you ever try to make a back-transformation of a Z-transformed expression ? Such things are common in communications engineering and the solution approach can get rather tricky (and the calculations can get rather long).

BTW: did anybody try this on a TI-92 ? I read somewhere, their Software is written by the Derive guys.


#7

Sure I've learned about integrals lot of years ago, so I can perform that type of operation by myself (with a pen and a blank sheet of course), the thing I'm questioning is about the weakness of the 48GX symbolic engine in this simple and common case. Just it.

I think the second part of the reply could be an answer...

Thanks


#8

I didn't want to create the impression you can't integrate. I just thought, the calculation would be a part of the answer.

What I wanted to say, is that these things can easily be rather complicated for a computer (especially for such a small device like the 48 with its limited ROM space).

I don't even know, if Derive uses the same method (compares pattern), just with a larger library (and a little better abstraction software).

My problem with the 48 (SX and GX) is the slow response to keyboard inputs sometimes, not the limited symbolic capabilitis.

I'd think, this integral is a rather complicated problem for a machine to solve (with substitution and subtraction).

#9

Go to www.hpcalc.org . There is software there to "boost" the performance of the HP48GX with respect to symbolic integration. I have little to no need for such in my job. I am more likely to need good numeric functions, which the 48GX has plenty of.


#10

If you really want to do symbolic integration on an HP48GX, you need additional software. Get ALG48 (including the optional INT library) and/or ERABLE from www.hpcalc.org. Both packages are free, although they consume so much memory that you may want to buy some RAM cards.

Both ALG48 and ERABLE can determine the indefinite integral of sin(x)*cos(x) in about 3-5 seconds.

The INTVX function of ERABLE returns (-1/4)*cos(2*x)

The INTGR function of ALG48 returns (-1/2)*cos(x)^2

The new HP49 has built-in calculus software based on ERABLE, so it should be able to handle this problem as well.

The TI-89 and TI-92 have built-in calculus software, which comes from the makers of DERIVE. In this case, the TI-89 returns -(cos(x)^2)/2.


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