This integral taught me Feynman's technique

This is an integral from the 2005 Putnam exam. It's the first integral I had ever solved using Feynman's trick of differentiating under the integral sign and I think it's one of the best examples on how to apply the technique.

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Advanced MathWear:

[emanuellandeholm]

I love when the derivative jumps over the integral sign and becomes a partial. Never gets old! :D

[thiagorc]

Great video. Finally i have understood the famous Feynman's technique.

[arunknown]

had my math exam today, and your channel really helped!

[baptFulbion]

I knew this one with the king's property "∫f(x)dx = ∫f(a+b-x)dx" with the same result! It's nice seeing the same problem resolved by 2 different approaches!

[SuperSilver]

I recently got introduced to a similar integral except the integrand is

Int(ln(1+x^2)/(1+x)) from 0 to 1. You should try this one with Feynman’s trick, the choice and placement of parameter might surprise you.

[anaymulay]

This problem was in our college exam but we were told to do it by trigo
Since by seeing your videos i tried it by Feynman and got answer really quick
Thanks bro

[intellix]

The integral / derivative switch up doesn't only require the integral to converge for certain values of alpha, normally you also want the partial derivative with respect to x to be continous by parts and the partial derivative with respect to alpha to be continous and smaller than an integrable function depending only of x between 0 and 1 (in that case x / (1+x²) wouldve worked)

[bandishrupnath]

Just wonderful integral and its explanation ❤sir.

[mystik]

i think a simple tanx = u, then applying integral (f(x) = integral f(a+b-x) will work where the limits are a to b

[HenkVanLeeuwen-io]

Usually log is the base 10 logarithm and ln is the natural logarithm.

[MrWael]

Thank you for your featured effort. Instead of log/2, it should be log2/2.

[bobbybannerjee]

Brilliant.
Can you tell us what's the name of the software/app into which you write ✍️ so that it shows up on your computer screen?

[joshuaiosevich]

I tried defining I(a)=integral from 0 to 1 of (x+1)^a/(1+x^2) dx, since I’(a=0) gives the integral in this video, alas the resulting integral is no better than what we started with, or at least I got stuck on it!

[CaioFalconieriLima]

Is possible to integrate 1/(1+x^5) with thia method?

[Czeckie]

can I use this method for computing the integral with variable upper bound and parameter? I mean I(t, a) = integral of log(1+ax)/(1+x^2) from zero to t. I'm getting a formula, but it's numerically definitely wrong.
EDIT: I see where's the problem. The observation from 9:33 doesn't work in general, so I only computed I(a, a)=arctan(a)*log(1+a^2)/2 which is no good!

[adhamkassem]

Great solution development, Please solve Bee integrals in the next videos

[giuseppemalaguti]

L'ho fatto con le serie risultato è corretto, ma non ho voglia di raggrupparli...

[quite_unknown_]

Could you solve this with 1/(1+x^2) = sum_(k=0) (-1)^k x^{2k} and then switch sum & integral, and then you get x^(2k) * log(1+x), solvable with IBP?

[skyethebi]

3:54 since when did partial fractions feel hospitable lol

[Mr_Mundee]

just do tan sub and some manipulation (no feynman)