
Did you know that there's an algorithm for symbolic integration that always works? - amichail
http://en.wikipedia.org/wiki/Risch_algorithm
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cousin_it
Hmm. The article says it's not really an algorithm, because one of the steps
requires checking if an expression is equivalent to zero, a problem that isn't
today known to be decidable. Anyone here knowledgeable enough to supply
details?

Edit: found the details in amichail's link. Yes, it's true; the algorithm
isn't complete as of today.

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amichail
The algorithm will find the answer if one exists or tell you that there is no
answer possible.

I wonder why it's never mentioned in high school Calculus classes. Symbolic
integration as presented there looks very magical -- not algorithmic at all.
You might mistake it for an AI problem (which it was for a while).

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lacker
Learning a complicated algorithm to deterministically answer a question
doesn't really help you understand the subject better in your first year of
calculus. I'd rather have classes focus on different ways of understanding the
simple parts of the subject, like the different applications of integrating
polynomials or simple trig functions. Or even simpler stuff, like ways to
think about functions other than a graph of x versus y. Or other ways to
intuitively understand why d(f(2x)) = 2f'(2x). Or the relation between
integration and big-O notation.

An intuitive understand of integration by parts is more valuable for a math
student than this algorithm. (Although I doubt most high school calculus
students could explain integration by parts a year later.)

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amichail
_An intuitive understand of integration by parts is more valuable for a math
student than this algorithm. (Although I doubt most high school calculus
students could explain integration by parts a year later.)_

Symbolic integration has a "problem solving" feel to it in high school and
teachers generally focus on this part -- making many students feel stupid in
the process if they can't figure it out.

Why not just tell them there's an algorithm for it and allow them to use a
program to do the symbolic integration whenever necessary while solving an
applied problem?

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lacker
I would say that learning "problem solving" is more important than learning
how to type a formula into integrals.wolfram.com (which you should certainly
use if you have an actual applied problem). Plus, if you want to go on to
multivariable calculus you will need to understand the principles rather than
just being able to get an answer to an integration.

I agree that calculus seems too "magical". But we should solve that by
explaining more of the intuition and heuristics behind solving calculus
problems, rather than teaching people to solve problems by brute force.

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amichail
There's plenty of problem solving left in the applied stuff.

And not everyone who takes Calculus in high school goes on to learn
multivariable calculus. High school Calculus should be more applied IMO.

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sp332
Guys, stop downvoting posts just because you disagree with them. If you have
something to add to the discussion, add it. Parent was not abusive or off-
topic or otherwise deserving of downmodding.

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sketerpot
Another disappointing limitation that nobody has mentioned yet: this only
works for elementary functions with integrals that are elementary functions.
If you try to get this thing to integrate exp(x^2) dx, it will (correctly) say
that there is no elementary integral. But the integral _exists_ , and it's
quite possible to find an infinite series solution, or to define the solution
in terms of the erf function.

So, this is cool but not all it's cracked up to be.

[http://en.wikipedia.org/wiki/Elementary_function_(differenti...](http://en.wikipedia.org/wiki/Elementary_function_\(differential_algebra\))

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amichail
You could define a new function to represent the answer where there would be
none otherwise, but that's not particularly useful unless that function is
important in numerous other contexts.

And btw, the algorithm has been extended to handle some non-elementary
functions.

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amichail
Also see
[http://groups.csail.mit.edu/mac/users/gjs/6.945/readings/sim...](http://groups.csail.mit.edu/mac/users/gjs/6.945/readings/simplification/moses-
stormy.pdf)

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vonsydov
Yeah lol.

