This integral is ridiculous.

Oh the satisfaction. We got Feynman's trick, funky substitutions, double integrals and...well....boring old partial fractions all wrapped up in one integral.

0:00 Method 1: Funky substitutions
4:15 Method 2: Double Integrals
9:57 Method 3: Feynman's Trick

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[rzeqdw]

10:30

Reminds me of an old math joke I learned in college

A mathematician and an engineer are working on math homework together involving 7-dimensional mathematics. The engineer is struggling to really understand the subject matter, but the mathematician is zipping through it

The engineer says "how is this so easy for you? I mean, 3 dimensions is easy, but 7? I just can't wrap my head around it"

The mathematician replies: oh it's easy. I just think of it like an N-dimensional problem, and set N to 7"

[TomFarrell-pz]

I like how you explain why you choose each step. The less mystery the better education!

[journeymantraveller]

So impressive. Use of basic identities and well known rules. Adding the integrals and logs.

from Morocco thank you for this clear complete full proofs

[VibratorDefibrilator]

There are math channels on youtube where this video would be three times longer. I hate long and unnecessarily detailed explanations of supposedly obvious techniques. This video is a nice exception, the right amount of (giant) steps a mathematician needs to take to solve a problem, any problem. The emphasis should be on the ideas, not the techniques to implement them.

I followed everything with ease, although I can't figure it out off the top of my head. Now I know.

Thank you, the recreational mathematics is the best.

[mnek]

This question is question A5 from the well-known Putnam (2005) Math Competition!

[anthonyheak]

Very nice explanation professor! Thank you!

[druminusr]

That’s really cool I was thinking tabular integration at the start did not see the trig sub coming.

[moustaphamoustapha]

It can also be solved by placing x=(1-t)/(1+t)

[LojKam]

Thank you very much. This enriched my thought and helped me solving a lot of integrals. Please make more videos about solving way more integrals it's literally cool and I'm learning ❤

[dentonyoung]

The first method was the simplest for me to understand.

[thisisntthefirsttime]

great video! there's also another method involving some rather tedious work but also works for Serret's Integral

1. substitute x = tan(theta)
2. cancel out sec^2
3. combine the inner term of the natural log
4. utilize the harmonic addition for its numerator
5. utilize log properties twice to get three terms (the first term directly gives the final answer)
6. cancel out the remaining cosine terms using King's integral

[Kobe]

Do you have a video on your set-up?
What softwares and set up are you using for this video?

[nanamacapagal]

The way I solved it (method 4): I expanded ln(1+x) into its Taylor series.

Took way longer and had to use a lot more tricks. The integrals themselves became quick and easy (albeit infinitely many), but then there was the summation part.

Had to use some weird manipulations, and ended up using these two specific sums over and over again
1/2 - 1/4 + 1/6 - 1/8 + ... = ln(2)/2
1 - 1/3 + 1/5 - 1/7 + ... = pi/4

What matters is at the end I got to pi * ln(2) / 8. So it's all good

[slavinojunepri]

The denominator screams a geometric series with -x^2 as the common ratio, which absolute value is less than 1.

[richardheiville]

The easiest method is not used in this video. The change of variable u=(1-x)/(1+x) is the best way to compute this.

[Fiery-kr]

I solved this same question in my high school finals. Putting x=tanθ! ❤❤❤

[TryHardNewsletter]

In method 3 you do I(a) = integral from 0 to 1 of ln(1+ax)/(1+x^2)dx and find I(1). What if you also put "a" in the bounds of the integral like I(a) = integral from 0 to a of ln(1+ax)/(1+x^2)dx and find I(1)? This is probably harder, but there might be some nice cancellations. There's a chance.

[Happy_Abe]

@6:04 doesn’t continuity only guarantee this in the case of a bounded interval, but in general we would need the integrals to be integrable to do the swap?

[Fiery-kr]

It was the same problem i solved in my high school finals. Just putting x = tanθ.. ❤❤