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Learning Feynman's Trick for Integrals

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Re: Learning Feynman's Trick for Integrals

#11

Post by impossiblefork »

owalt wrote: post #5
impossiblefork wrote: post #2

It starts off with a pretty major error. I'(t)=\int_0^1 \partial/(\partial t)((x^t - 1)/(ln x))dx = \int_0^1 x^t dx=1/(t+1), when it is actually equal to \int_0^1 x^{t-1}/ln(x)dx. These two are definitely not always equal to each other.

No, it is correct. The integral is with respect to x, and the ordinary/partial derivatives are with respect to t. Written out fully, the derivative computation is d/dt (x^t - 1)/ln(x) = d/dt [exp(ln(x)t) - 1]/ln(x) = ln(x)exp(ln(x)t)/ln(x) = exp(ln(x)t) = x^t. Edit: d/dt exp(ln(x)t) = ln(x)exp(ln(x)t) by the chain rule, while d/dt (1/ln(x)) = 0 since the expression is constant with respect to t. There are convergence…

Ah, yes. I don't understand how I differentiated with respect to x instead of t, but...

Re: Learning Feynman's Trick for Integrals

#12

Post by biophysboy »

> So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's

This is the most important lesson I learned in grad school. Methods are so important. I really think it is the core of what we call "critical thinking" - knowing how facts are made.

Re: Learning Feynman's Trick for Integrals

#13

Post by measurablefunc »

zeroonetwothree wrote: post #9

It’s interesting he mentions he doesn’t like contour integration since many integrals can be done either way. Feynman’s trick is equivalent to extending it into a double integral and then switching the order of integration.

Don't forget to check for the necessary measurability & integrability of the sections (f(a, y), f(x, b)) before switching the order: https://en.wikipedia.org/wiki/Fubini%27s_theorem?useskin=vec....

Re: Learning Feynman's Trick for Integrals

#14

Post by analog31 »

I don't know if this is exactly the same as what I learned in high school as "integration by substitution."

A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "form" of a problem and applying an algorithm appropriate for that form, drawn from the most recent chapter.

In the TFA, the natural log in the integrand was a dead give-away because it only comes from one place in the standard order of topics in calculus class.

Is this what we call intuition?

The students called this the "trick." Many of them had come from high school math under the impression that math was subjective, and was a matter of guessing the teacher's preferred trick from among the many possible.

For instance, all of the class problems involving maxima and minima involved a quadratic equation, since it was the only form with an extremum that the students had learned. Every min/max problem culminated with completing the square. I taught my students a formula that they could just memorize.

The whole affair left me with a bad taste in my mouth.

Re: Learning Feynman's Trick for Integrals

#15

Post by [deleted] »

analog31 wrote: post #14

I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "for…

[deleted]

Re: Learning Feynman's Trick for Integrals

#16

Post by dunham »

zeroonetwothree wrote: post #9

It’s interesting he mentions he doesn’t like contour integration since many integrals can be done either way. Feynman’s trick is equivalent to extending it into a double integral and then switching the order of integration.

This reminds me of the "snake oil method" for generating functions. It's been many years, but I remember it as adding another sigma and then swapping them.

Re: Learning Feynman's Trick for Integrals

#18

Post by BearOso »

lordnacho wrote: post #10
zeroonetwothree wrote: post #8

Earlier quoted context omitted.

That’s how most of math works past high school. It requires a lot of practice and intuition.

I don't know about this particular case though, I get the feeling there's a system to it that can be exploited by eg Wolfram. It's just that you're in the dark for a long time before you find the switch.

I think it just tokenizes everything and does pattern matching to find compositions it can exploit. It's not unlike compiler optimization.

Re: Learning Feynman's Trick for Integrals

#19

Post by cyberax »

lordnacho wrote: post #10
zeroonetwothree wrote: post #8

Earlier quoted context omitted.

That’s how most of math works past high school. It requires a lot of practice and intuition.

I don't know about this particular case though, I get the feeling there's a system to it that can be exploited by eg Wolfram. It's just that you're in the dark for a long time before you find the switch.

Your intuition is right. There is a general algorithm for finding the antiderivatives: https://en.wikipedia.org/wiki/Risch_algorithm Its simplified form can solve pretty much all the undergrad antiderivation problems.

I'm a math major, but I consider the time spent learning the tricks for antiderivation to be kinda useless.

Re: Learning Feynman's Trick for Integrals

#20

Post by charcircuit »

analog31 wrote: post #14

I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "for…

I think it's intuitive to assume what you are being tested on is what is being taught by the book or the teacher. It's unfair otherwise.
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