I used what we proved about α and β as inspiration to solve the integral in a pretty nice way : * Take x = sqrt(1+θ²) + θ Then similarly to the case with α and β, we have that x - 1/x = 2θ. * Under this change of variable, α and β clearly become -π/6 and π/6 respectively. * dx becomes (θ/sqrt(1+θ²) + 1)dθ, however since cos(2θ)*θ/sqrt(1+θ²) is odd, that part vanishes in the integral (the bounds are opposite) * we are left with integral of cos(2θ)dθ from -π/6 to π/6, which gives sin(π/3). Nice video as always!
Very creative! At first I tried doing this through u-sub but soon realised that it was going to be an extraneously long solution, problem being absence of 1 + 1/x^2 term. Although I think this is the best method, if one were to go through the efforts of doing the u-sub completely, then I suppose IBP would've eventually yielded the solution. Just as a back up brute force plan.
Reminded me of this one: ruclips.net/video/pOCisqitZbk/видео.html Note that until the very last step only the evenness of cos() and properties of the bounds of integration are used here, so that would apply to any integral over f(x+1/x) from a to b with ab=1 and f even.
This reminds me of calc II test in college. It was the ridiculously ugly integral, just really painful. After all the work and effort it simplified out to be just 1+1 =2. Pretty sure the professor was just having a laugh at our expense.
I did this with a x = e^t substitution since the (x - 1/x) argument to cos reminded of e^t - e^{-t} which is 2sinh(t), then added and subtracted integral on the same log-transformed bounds of the integral of e^{-t} * cos(e^t - e^{-t}) dt. Then I briefly got stuck but the hint came in helpful with the "extra" definite integration of the e^{-t} * cos(e^t - e^{-t}) dt term which is just a negative copy of the original integral, leaving 2 * I(t) = integral from -ln(beta) to ln(beta) of (e^t + e^{-t}) * cos(e^t - e^{-t}) dt, which is an easy u-sub.
Your engagement by commenting on that micro-error has boosted this channel in YT's algorithm. If you haven't noticed that it's not a bug, it's a feature to get extra engagement, then you haven't really caught on to what running a YT channel is all about. Put aside the annoyance of these 'mistakes' and just enjoy the channel for the great coverage of topics.
@@Hiltok Are you seriously suggesting the commenter should enjoy this channel but *not* boost it? That's plain silly. If everyone involved enjoys the content it makes no sense to discourage engagement. 🤔
@@winteringgoose I'd rather people engage with the maths than remaining upset by trivialities like whether the thumbnails exactly match the content. There have been multiple comments about it over time so if it was going to be 'fixed' that would have happened by now. It hasn't, so one can infer a choice has been made. [I used to pause the videos to write comments about transcription errors in Michael's work. Over time, I learned that he always corrects them later on, either in person on the video or by superimposing script over the offending bits. It happens often enough that it's 'just how he rolls'.]
@@angelmendez-rivera351 My comment was based on drawing an inference from observed behavior. What reason do you ascribe to Michael and his crew ignoring the issue of thumbnails not exactly matching the problem at hand, about which there have been multiple comments over time? If it is not seen as a triviality that leads to extra comments/engagement to feed the YT algorithm, then it suggests they lack responsiveness to their viewers wanting greater accuracy. I know I'd prefer to think of the people running this channel as smart enough to know how YT works and looking to improve outcomes for the channel. I don't think I would want to categorize them as uninterested in the concerns of their viewers and repeatedly slipshod in their productions.
I'm new to integrals, why are you just allowed to substitute u with x to get the 1/x² integral? I can think of a quick proof for all the rest of the steps but i don't get that step, shouldn't it be replaced with 1/x, giving the original integral again?
Definite integrals that are the same except for the labelling of the variable are equal. You can suppose you have the antiderivative F(x)/F(u) and substitute the limits to prove this you will get F(beta)-F(alpha) in both cases.
There is no x² anywhere in that integral which could cancel the x^-2. What are you talking about? He essentially is only renaming the dummy integration variable from u to x.
I have a feeling that this problem was constructed going backwards. It is fun, but very artificial. Speaking as someone who studies physics -- you will almost never encounter something like this in "real life".
Yes, the bounds or limits of integration are very artificial or very special, or cooked up, in order to solve the integral. I think the indefinite integral is very hard to find or may not be an elementary function.
Further, the new x and the old x are not the same but that's ok when dealing with definite integrals. When all is said and done we are ending with a constant after evaluation.
Seriously, can anyone solve this integral question at first sight? There are so many tricks here. In my opinion, even a calculus teacher would take some long time to find all the tricks in this question. ( I wonder whether my assumption is right🙄🙄)
It's easier than it appears because of two factors: 1) substituting u = 1/x is quite natural; 2) the integration bounds are so specific that serve as hints basically for point 1)
I looked at the screen. The integral function cos(1-1/x) is depicted. I paused in watching the video and started to solve. Got the result. I decided to compare. It turned out that the integrand function cos(x-1/x). Complete shit and lack of respect for the audience.
Not the first time the thumbnail was wrong, though it seems to be changed now. I do wonder what's going on because Michael is the only math guy I follow that has recurring instances of mismatched thumbnails at time of upload.
I think that was an honest mistake by him- part of me thought it was the equivalent of a mobile game ad where the ad is fake just to lure you in, but the fact that he corrected it makes me think it was just a mistake
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I've noticed myself getting better at integrals just by watching Michael Penn videos. I was able to solve this one is under 5 minutes
such a fun integral! loved how it combined with the 1/x^2 factor! great video michael keep it up :D
I used what we proved about α and β as inspiration to solve the integral in a pretty nice way :
* Take x = sqrt(1+θ²) + θ
Then similarly to the case with α and β, we have that x - 1/x = 2θ.
* Under this change of variable, α and β clearly become -π/6 and π/6 respectively.
* dx becomes (θ/sqrt(1+θ²) + 1)dθ, however since cos(2θ)*θ/sqrt(1+θ²) is odd, that part vanishes in the integral (the bounds are opposite)
* we are left with integral of cos(2θ)dθ from -π/6 to π/6, which gives sin(π/3).
Nice video as always!
I would have majored in Math if I had seen Michael Penn's videos growing up in the 80s and 90s.
I think he himself would have been happy to see those videos these years when he was growing up too :)
Very creative! At first I tried doing this through u-sub but soon realised that it was going to be an extraneously long solution, problem being absence of 1 + 1/x^2 term. Although I think this is the best method, if one were to go through the efforts of doing the u-sub completely, then I suppose IBP would've eventually yielded the solution. Just as a back up brute force plan.
But he did ubsib in tbr video..was yours different
I absolutely love the way you present your videos!
Wow, wow, wow.
Great, if contrived, problem.
Thank you, professor.
Reminded me of this one: ruclips.net/video/pOCisqitZbk/видео.html
Note that until the very last step only the evenness of cos() and properties of the bounds of integration are used here, so that would apply to any integral over f(x+1/x) from a to b with ab=1 and f even.
great video as usual. I'm already a patreon supporter and I'm super happy that we are getting closer to the 1k/mo goal!
It's a joy to watch!
I wasn't really sure where he was going with this but then at 7:49, it all clicked, and it's a very slick solution I might add!
I love this integral
Well that was a treat
ok that took a twist I wasn't expecting. fun!
This reminds me of calc II test in college. It was the ridiculously ugly integral, just really painful. After all the work and effort it simplified out to be just 1+1 =2. Pretty sure the professor was just having a laugh at our expense.
College? I am doing calc 2 in ninth grade.
Brilliant!
Beautiful!
I did this with a x = e^t substitution since the (x - 1/x) argument to cos reminded of e^t - e^{-t} which is 2sinh(t), then added and subtracted integral on the same log-transformed bounds of the integral of e^{-t} * cos(e^t - e^{-t}) dt. Then I briefly got stuck but the hint came in helpful with the "extra" definite integration of the e^{-t} * cos(e^t - e^{-t}) dt term which is just a negative copy of the original integral, leaving 2 * I(t) = integral from -ln(beta) to ln(beta) of (e^t + e^{-t}) * cos(e^t - e^{-t}) dt, which is an easy u-sub.
Why not just start with cosine difference formula isn't that simpler and more obvious?
I had the "aha!" at 4:13 A very funny integral !
We can use glasser master theorem if the bounds are between -infinity to +infinity
What a lucky coincidence that all that crazyness boils down to such a neat result in the end ;-)
almost as if it was intentionally constructed to be se
@@chri-k It was intentionally constructed to be so
I second that.
the trick works fine too for computing integral of 1/(1+x^4) x=0,infinity
Waou!1 amazing strategy. Starting my breakfast with this Pb.
Fun integral
Michael or his editor should be more careful with the thumbnails, the problem in the video isn't the same as the problem in the thumbnail
Your engagement by commenting on that micro-error has boosted this channel in YT's algorithm. If you haven't noticed that it's not a bug, it's a feature to get extra engagement, then you haven't really caught on to what running a YT channel is all about. Put aside the annoyance of these 'mistakes' and just enjoy the channel for the great coverage of topics.
@@Hiltok Are you seriously suggesting the commenter should enjoy this channel but *not* boost it? That's plain silly. If everyone involved enjoys the content it makes no sense to discourage engagement. 🤔
@@winteringgoose I'd rather people engage with the maths than remaining upset by trivialities like whether the thumbnails exactly match the content. There have been multiple comments about it over time so if it was going to be 'fixed' that would have happened by now. It hasn't, so one can infer a choice has been made.
[I used to pause the videos to write comments about transcription errors in Michael's work. Over time, I learned that he always corrects them later on, either in person on the video or by superimposing script over the offending bits. It happens often enough that it's 'just how he rolls'.]
@@angelmendez-rivera351 My comment was based on drawing an inference from observed behavior.
What reason do you ascribe to Michael and his crew ignoring the issue of thumbnails not exactly matching the problem at hand, about which there have been multiple comments over time? If it is not seen as a triviality that leads to extra comments/engagement to feed the YT algorithm, then it suggests they lack responsiveness to their viewers wanting greater accuracy. I know I'd prefer to think of the people running this channel as smart enough to know how YT works and looking to improve outcomes for the channel. I don't think I would want to categorize them as uninterested in the concerns of their viewers and repeatedly slipshod in their productions.
So satisfying!
I'm new to integrals, why are you just allowed to substitute u with x to get the 1/x² integral? I can think of a quick proof for all the rest of the steps but i don't get that step, shouldn't it be replaced with 1/x, giving the original integral again?
Definite integrals that are the same except for the labelling of the variable are equal. You can suppose you have the antiderivative F(x)/F(u) and substitute the limits to prove this you will get F(beta)-F(alpha) in both cases.
Cool integral simplification tricks 🥵
okay, it was epic!
1/s ds /x
Woah!
If you go back to “x“, should it not be “x^2“ instead of the inverse?
arithmic propertiEas
wolfram gives some crazy results when ∫ (1+1/x^2 ) cos(x - 1/x) dx
Same integral 0 to infinity
the argument of that integrand is not quite a nice place to stop... maybe add a 1/x^2 term to 1-1/x
.?.?.?
I cant even integrate the classification of all finite simple groups over the unreal numbers
This looks like the cauchy smolich transform.
WHO came up.woth this integral to begin with and WHY would they??
i did three substitutions to come to that even function argument switch 🤦♂🤦♂
clever solution, thumbnail is wrong btw
I'm not seeing the step at 6:23. It seems like the x^2 should cancel the x^-2, yielding the original integral.
There is no x² anywhere in that integral which could cancel the x^-2. What are you talking about?
He essentially is only renaming the dummy integration variable from u to x.
I have a feeling that this problem was constructed going backwards. It is fun, but very artificial. Speaking as someone who studies physics -- you will almost never encounter something like this in "real life".
You can do this by a less clever method too, and the problem itself is also very non specific and simple to state
Yes, the bounds or limits of integration are very artificial or very special, or cooked up, in order to solve the integral.
I think the indefinite integral is very hard to find or may not be an elementary function.
but when you do, you now know how to solve it
@@jesusandrade1378 but the only thing relevant about the bounds is b-a=π/3 and ab=1
whoa !
6:25 I don't understand how we can substitute u=x back into the integral when u was previously defined as 1/x. Can someone please explain? Thanks
Variable names don't matter
he just re-named u to x
Further, the new x and the old x are not the same but that's ok when dealing with definite integrals. When all is said and done we are ending with a constant after evaluation.
I don't get the even function argument, but I guess it is simple enough...
👍
Oh hi
Why not use cosine difference formula tonsokve. Isnt this SCREAMING gor youbto do that? Dodnt any onebekse do this??
👍👍👍👍👍👍
Did he really look up sin of 60? 10:14
no, he know it
insane :DDDD
Seriously, can anyone solve this integral question at first sight? There are so many tricks here. In my opinion, even a calculus teacher would take some long time to find all the tricks in this question. ( I wonder whether my assumption is right🙄🙄)
It's easier than it appears because of two factors: 1) substituting u = 1/x is quite natural; 2) the integration bounds are so specific that serve as hints basically for point 1)
Oh, your point makes sense! Maybe I should be more used to wider range of problems. Thanks!!
🌹
barely complicated
I looked at the screen. The integral function cos(1-1/x) is depicted.
I paused in watching the video and started to solve. Got the result. I decided to compare.
It turned out that the integrand function cos(x-1/x).
Complete shit and lack of respect for the audience.
Not the first time the thumbnail was wrong, though it seems to be changed now. I do wonder what's going on because Michael is the only math guy I follow that has recurring instances of mismatched thumbnails at time of upload.
The picture was imperceptibly corrected. and my comment turned into not adequate.)))
I think that was an honest mistake by him- part of me thought it was the equivalent of a mobile game ad where the ad is fake just to lure you in, but the fact that he corrected it makes me think it was just a mistake
Cute!
cute!
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