[HN Gopher] Learning Feynman's Trick for Integrals
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       Learning Feynman's Trick for Integrals
        
       Author : Zen1th
       Score  : 60 points
       Date   : 2025-11-29 19:55 UTC (3 hours ago)
        
 (HTM) web link (zackyzz.github.io)
 (TXT) w3m dump (zackyzz.github.io)
        
       | impossiblefork wrote:
       | 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.
        
         | owalt wrote:
         | 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 considerations that were not discussed in
         | the blog post, but the computations seem to be correct.
        
           | impossiblefork wrote:
           | Ah, yes. I don't understand how I differentiate with respect
           | to x instead of t, but...
        
       | lordnacho wrote:
       | My issue with both this and u-substitution is that you don't know
       | what expression to use. There are a LOT of expressions that
       | plausibly simplify the integral. But you have to do a bunch of
       | algebra for each one (and not screw it up!), without really
       | knowing whether it actually helps.
       | 
       | OTOH, if I'm given the expression, it's just mechanical and
       | unrewarding.
        
         | zeroonetwothree wrote:
         | That's how most of math works past high school. It requires a
         | lot of practice and intuition.
        
           | lordnacho wrote:
           | 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.
        
       | tacitusarc wrote:
       | I just finished Mathematica by David Bessis and I wish this
       | information was presented in the way he talks about math: using
       | words and imagery to explain what is happening, and only using
       | the equations to prove the words are true.
       | 
       | I just haven't had to use integral calculus in so many years, I
       | don't recall what the symbols mean and I certainly don't care
       | about them. That doesn't mean I wouldn't find the problem domain
       | interesting, if it was expressed as such. Instead, though, I get
       | a strong dose of mathematical formalism disconnected from
       | anything I can meaningfully reason about. Too bad.
        
         | chrisweekly wrote:
         | That's one of the things I like best about
         | https://betterexplained.com -- it focuses on ways to gain
         | intuition about a given math concept, using visuals and
         | metaphors as appropriate. If only math education were always
         | presented like that....
        
       | zeroonetwothree wrote:
       | 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.
        
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       (page generated 2025-11-29 23:00 UTC)