BIG brilliant integral

[@debasishphukon5345]

"dont forget the dx, otherwise people will sue me" 😂😂😂😂😂😂😂😂 i died 😂😂😂

[@dreamshack2053]

😂😂😂😅

[@charlesleninja]

What i love about maths is when you find a crazy way to solve something, and the answer comes out to be something appenrently unrelated. Like how you get pi by integratinf a bunch of logaritms.

[@manishkumarsingh3082]

I substituted x=1/y and it worked :D

[@Morphysince94]

How can i tackle the extra -1/y²dy from the derivative.. also should we interchange the newly generated limit from Y= (infinity to Zero) to (Zero to infinity)? Sign change may come

[@Morphysince94]

I used x=tan (theta) and it worked, there are lot of ways that I didn't know

[@Morphysince94]

Oh I just realized your integration technique is actually the same as shown in the video, Blackpenredpen used the known function, and you came to that same integration form through x=1/y, this is much better if anybody wants to learn on their own

[@CheapPhysics]

It is as same as the video's method......

[@aurithrabarua4698]

'Brilliant ' website Brilliant Integral Brilliant Technique Brilliantly solved 😊😊😊😊

[@frankwei222]

I know it's three weeks late, but I'm doing a marathon to catch up! :) I paused the video to try to tackle the integral myself and I think I've found an alternate way to solve it. First, take the general case by replacing 11 with a and 3 with b. Next, using Feynman's Trick, differentiate with respect to a and substitute x=tan t. Through some crafty substitution, the general case is equal to pi/4(a-b). So set a=11 and b=3 to get 2pi as the answer. I wish I could be more precise, but I'm not sure how to type math symbols into KZbin comments...

[@gnikola2013]

Now that's what I call an interesting integral

[@yashpermalla3494]

Waiting for the 100k subs milestone, so excited! btw This video was awesome, as usual! i needed a refresher of integral calculus!

[@blackpenredpen]

yash permalla thank you very much. Yes I am waiting for that 100k too and hopefully it happens soon :)

[@GreenMeansGOF]

Since we are all adults, we say that the integral equals τ.

[@GeodesicBruh]

Yes

[@GeodesicBruh]

τ>π Literally

[@simenjorissen5357]

I'm 15, therefore we're not all adults and this is not allowed

[@user-qb5gw7tc9e]

Other way to solve: Consider x = exp(t) so dx = exp(t)dt x = 0 -> t = - infinity x = infinity -> t = infinity ln((1 + x^11)/(1 + x^3)) = ln (x^5.5) + ln(x^-5.5 + x^5.5) - ln(x^1.5) - ln(x^-1.5 + x^1.5) = ln(x^4) + ln((x^5.5 + x^-5.5)/(x^1.5 + x^-1.5)) = 4t + ln(ch(5.5t)/ch(1.5t)) (sum of odd and even functions) Since integral if odd function equals to 0 I is integral of 2dt/ch(t) (can be solved easily)

[@Roy-ub7uz]

i am a 12th grade student who is currently preparing for entrance exam. ive started watching your videos long back when i was probably at 9th and i count understand a thing but i loved the way u teach and do maths. now i myself can relate to mant of the math that you do and its just feels sooo good much love ;)

[@WilliamLeeSims]

I feel like there should be some analysis around 1 to make sure the function is well behaved. (Spoiler: It is.)

[@SimonClarkstone]

Yeah I was thinking this isn't well behaved around x=1 but does seem to just have a point missing, rather than an asymptote.

[@bilz0r]

I'm grateful for your videos, but I much prefer it when you do it on the whiteboard!

[@hroseman]

This one is a setup. More interesting is the process in defining the integral. Guaranteeing that a complex expression simplifies so thoroughly.

[@holyshit922]

My way is to divide interval into [0;1] and [1;infinity] and in the integral on interval [1;infinity] substitute u=1/x then when you combine these two integrals you will have nice cancellations x=1 makes indegrand undefined and that suggest me such interval division

[@fengshengqin6993]

I have computed it in my head in 5 minutes and got the very same result with you :2Pi . But thanks for your sharing .

[@caesarinchina]

开玩笑嘛😳

[@fengshengqin6993]

@@caesarinchina 真没开玩笑，有的积分就是可以心算的。

[@caesarinchina]

@@fengshengqin6993 那你真的好厉害。 加我微信还是FB 什么的。 我是老外呀。

[@fengshengqin6993]

@@caesarinchina Sorry ！ I saw you had an Pingyin ID , and then I thought you were a chinese . My bad. My wechat ID is qinfengsheng001 .

[@caesarinchina]

@@fengshengqin6993 Sorry for late reply, very busy lately but I will add you, your math is impressive and I am a huge mathematics / physics enthusiast

[@tariniprasadraj534]

We can apply Taylor's expansion and Laurent's series.

[@Craznar]

I'll link to this video when I want to define 'simplify' in future :)

[@sayantanmazumdar9371]

alternate solution using kings property sub x=tan(theta) so u will be left with integral from 0 to pi/2 of ln((1+tan^11(x))/(1+tan^3(x))/(ln(tan(x)) dx set that equal I so I=integral from 0 to pi/2 of ln((1+tan^11(x))/(1+tan^3(x))/(ln(tan(x)) dx use ur friendly kings property tan(pi/2-x)=cot(x)=1/tan(x) I=integral from 0 to pi/2 of ln((1+1/tan^11(x))/(1+1/tan^3(x)))/ln(1/tan(x)) dx simplifying =integral from 0 to pi/2 of - (ln((1+tan^11(x))/(1+tan^3(x)))-ln(tan^8(x)))/ln(tan(x))dx =integral from 0 to pi/2 of (8ln(tan(x))-ln((1+tan^11(x))/(1+tan^3(x))))/ln(tan(x))dx =integral from 0 to pi/2 of 8-ln((1+tan^11(x))/(1+tan^3(x))))/ln(tan(x))dx =8*pi/2-I =4pi-I hence I=4pi-I=> 2I=4pi=> I=2pi

[@rontiemens2553]

I thought this video was fantastic. One of the best you have done to date. VERY illuminating.

[@dank9427]

I realized the other day that sinx+cosx-sinx-cosx differentiates onto itself, so I got excited before I realized that I was just differentiating zero. :(

[@yerr234]

xD

[@karthikm1348]

I love this guy . His accent and the way he explains subtle tricks. Just wanna prActise more integrals .

[@mike4ty4]

Try this one suggested in a comment on one of your previous videos: int (x sin(x))/(1 + cos(x)^2) dx. (indefinite.) Wolfram gives a result in terms of De Jonquiere's Polylogarithm and it fills up an entire PAGE. :) However I suspect it is possible to write the solution more compactly if you keep track of similar terms.

[@bigfuss4135]

at 7:05 this means that integral (ln(1+x^11) - ln(1+x^3) )dx is exactly as - integral (ln(1+x^11) - ln(x^11) -ln(1+x^3)+ln(x^3) )dx then integral(2*(ln(1+x^11) - ln(1+x^3)) dx= integral(ln(x^11) - ln(x^3))dx Good. Grand. Great. Gosh!

[@General12th]

I love how enthusiastic you get at 11:05. You're like an airplane about to take off!

[@shadowbane7401]

love the f(1/x)/x^2 property

[@Morphysince94]

I solved the problem with u substitution, as X=tan(y) dx=sec²(y)dy, after applying that the limit would be from 0 to pie/2, then using the king property of integration, I finally got the answer.. lot of tan^○(pie/2 - x) happened

[@kshitijgaur9635]

Thank you, very nice explanation!

[@cristiano7533]

I love the way you use your pens.

[@mohandoshi153]

Absolutely Brilliant - bprp - you rock.

[@smitashripad9757]

U should hv done live. So we could also give our suggestions and learn more.

[@soultheft]

Whoa. OK, I have about ten Calculus books of various flavors, and none of them mention the relationship you point out in your note at 2:10. Can you suggest a book or two that might cover these sorts of techniques or derives these sorts of relationships? This one is incredible. I'm going to roll it around on various integrals over coffee this morning, but I'd love some book suggestions. I see it's covered on Brilliant. I am a subscriber with the blackpenredpen discount! Yeah! Thanks so much. -Phil

[@blackpenredpen]

Hi Soultheft, Thank you for your support! In fact, I really don't use books that often nowadays since the Internet offers way much richer resources (and for free, too). Like you mentioned, I got this from Brilliant.org as well, where I learned a lot of new techniques from. Other sites such as quora or stackexchange are awesome too.

[@leif1075]

@@blackpenredpen Question are you saying you wouldbt have thought of this yourself?? I don't see how anyone would..I hope you can PLEASE respond ..Thanks

[@ANUJKUMAR-oe7ct]

You Are really Great sir Ji. You Are not a simple man.You are mathe magician.

[@mohammedaayachi3828]

the solution is quite elegant, it all depends on that note you used, but what I'm wondering is where could we possibly find such monstrous integrals.

[@hiki2793]

Just here to share bit of definite integration techniques try substitute in 1/x where ever u see lnx and unusual limits

[@siddhantsrinivas5975]

Substituting x= tan y and then replacing y with (pi/2)-y, might be a much faster method

[@PatrickCharrault]

Back to the original integral; I have one question : as the integral goes from "0" to infinitiy, isn't there a problem with ln(0) first ?

[@victoraugusto1698]

But it's the "integral" of ln(x) from 0 to infinity. You are not just replacing values

[@Alians0108]

As shown when taking the Riemann sum, integrals don't actually plug in the first value IIRC.

[@pgseducation915]

As it is improper integral then these type of limits are possible.

[@mallakbasheersyed1859]

This is an improper integral If it is discontinuous at some point we use to take limit at that point

[@rabindranathghosh31]

this is a pretty interesting limit: what is the limit as n approaches infinity of [(1 + 1/n)(1 + 2/n)...(1 + n/n)]^(1/n) did you figure out the answer?

[@rabindranathghosh31]

answer is √e

[@vikasbalani4310]

You take log of it and write it in summation form. As n tends to infinity it converts to a definite integral.

[@shrihari154]

Legends says that this legend is a legendary teacher.

[@mcwulf25]

Love this. But isn't there a "-" in your f(1/x) substitution? Confess I am watching this on silent so maybe you have said something on this. Really loving your videos.

[@yahiazakarialadhem9411]

thank you

[@derek3719]

Thank you for your videos! :) I like how you take on tough integrals I have never seen before and you provide such clear instruction, making it possible for amateur mathematics such as myself, easy to understand. I am not even a math or science major, graduated years ago in social work. Somehow I got addicted to math. At any rate, I went to that Brilliant.org and tried to solve integral (bounded by infinity to 0) of ln(2x)/(1+X^2). There was a step where they go from ln(2x)+ln(2x^-1) to 2ln2 and I am completely befuddled as to how they got this step. I tried natural log properties and U substitution and am lost. At some point could you do a video on this one or a couple more videos using this style of integration?

[@calvinlin5753]

Hint: ln a + ln b = ln ab. Apply this twice.

[@blackpenredpen]

Hi Derek! Thank you for your comment. I will think about that integral and can totally do more similar integrals in the future!

[@calvinlin5753]

To clarify, the problem is an example which is listed under the Integrations Trick - Inversions wiki.

[@derek3719]

Awesome, thank you!

[@derek3719]

Thank you Calvin, this was helpful. Silly me, I was applying the ^-1 to the (2x) instead of just the (x).

[@bruno68berretta53]

Bonjour. Mes compliments. (Complimenti dall'Italia).

[@cosmicvoidtree]

It’s funny how often we try to guess what things will end up being, and you just get tau

[@albertodelaraza4475]

Definitely a 'WOW!' moment when all those ln terms cancelled out! That Integral was a full notch above challenging. It got me thinking though: You have previous videos on how to create certain math problems, (to torture your beloved students!), and make sure it comes out with a nice answer that students can have a sense of satisfaction after solving it. How in the world did someone at Brilliant come up with this monster Integral and made sure it came out with such a nice answer like 2pi? Just creating such an Integral seems like an even greater challenge. What do you think? How would you have created a similar monster? ADLR (nom de KZbin)

[@blackpenredpen]

Alberto DeLaRaza ahhh!!! Thank you for your comment and question. Sounds like you enjoyed my previous videos on creating problem for students. And you got me thinking about this one. I do know have a good response yet. When I do, you will see a video.

[@krishnanadityan2017]

Brilliant !!!

[@scottwilliams7672]

Vraiment bien! C'est excellent!

[@blackpenredpen]

Thank you!

[@NoName-mr6go]

Very good video however ln(x) is zero at x=1 so you need to check that the integral converges at x=1

[@berkicy]

Just assume that you find the 4x but you forget to divide into 2 xD

[@nepasdisponible]

I would've definitely missed that.😂

[@nediadarth4999]

you made me love math 😍😍

[@blackpenredpen]

Thanks!

[@neilgerace355]

Hi bprp, how long ago did you reach 1M subscribers? I must've missed that event. Did you get an award from KZbin?

[@TN6625]

That was really brilliant! But I don't think I've seen that f(1/x)/x^2 before. BTW: On first seeing your work I thought you were a High School teacher with an integral hobby, but you must be a PHD prof. in some university like your associate Dr. Peyman. Is that right?

[@kqnrqdtqqtttel1778]

Beautiful!!!!!!

[@martonnemeth236]

Grande Inregralle Brilliante

[@surajpati5193]

Pls solve this... Min and max value of sinx +cosx +tanx + cosecx + secx + cotx...

[@prakhardwivedi3649]

Max value = ∞ Min value = -∞

[@surajpati5193]

How....

[@Aldron6]

approaching pi/2 from the left you have 1+0+infinity+1+infinity+0 = infinity approaching pi/2 from the right you have 1+0-infinity+1-infinity+0 = -infinity

[@surajpati5193]

I think it was min positive value...

[@Aldron6]

at x=0+ the function tends to infinity, and it tends to infinity at pi/2, then as all trig functions are continuous in that interval, so is the sum of them. Hence, there is a minimum in (0,pi/2). differentiate and set equal to 0, then express everything in sins and cosines and you get cosx-sinx +(1+sinx)/cos^2(x) - (cosx+1)/sin^2(x) = 0. The LHS is anti-symmetric in sin and cos, so if we can find an x such that sinx=cosx, the equation will be satisfied. then we know pi/4 to be such a value. evaluating the original function at pi/4 gives 2+3sqrt2

[@CFneo98]

would you look at that, the answer is a full rotation, i’d love to see a graph explanation if this formula

[@khaledab0mhmad667]

Ispeak english able solve. تكامل ln(×)dx/ln(×+1)

[@aboutmath2995]

Since lnx is not continuous at 0 how you integrate from 0 to infinity?

[@eliaschavez364]

it should be the limit when a---> 0 from the right side, integrating from "a" to the infinity but anyway it's almost the same

[@jemcel0397]

This integral is really brilliant. Wish signing up is free :(

[@calvinlin5753]

Signing up is free :)

[@shivamgtx4891]

Nice sir... You are best.. 😊😊😊😊

[@mihirchawla890]

bro u can also substitute x=tanu and then use property of definite integral that will be much easier solution

[@blackpenredpen]

mihir chawla and u got the same ans?

[@mihirchawla890]

yes

[@blackpenredpen]

mihir chawla Good job!

[@mihirchawla890]

thank you Sir!!

[@testonly4515]

Which property

[@infinitamo]

You should take on elliptic integrals analytically

[@tringuyen121]

IS THAT DORAEMON'S OPENING THAT I JUST HEARD!!!!

[@blackpenredpen]

tri nguyen yes

[@alejrandom6592]

This is art

[@qilinxue989]

Obligatory like before watching.

[@ernestschoenmakers8181]

Shouldn't you write int of f(x)*dx= - int of f(1/x)*dx/x^2 between 0 and infinity?

[@bod5236]

Best channel

[@jennytan7656]

Now that’s BRILLIANT!

[@hatembahri4314]

all i can say just amazing

[@abathur5011]

Beautiful

[@_DD_15]

If you do a sub x=1/y, f(x)=f(1/y), dx=-2dy/(y^2). By renaming y into x, (-1/2)f(1/x)dx/(x^2). Where is the -1/2 in the formula at 2:42 ? Or I'm missing something?

[@lalitdogra22]

Can you please make a video on how to ascertain whether a number is prime or not just by looking at it? All i know is that prime numbers are of the form 6n±1 but I can't be sure always whether it's the (6n+1)th number or (6n-1)th number (or both). This uncertainty annoys me a lot when I see an odd numbered registration plate/house number :-(

[@Alex-sz5zd]

Why didn’t you chang scope when you use 1/x replace x?

[@spencergee6948]

Who on earth would nee to integrate such an obscure integraol? Does it represent some important physical phenomenon? Even though it is clever maths, what is the point?

[@chetnabudhraja3253]

I can't believe I solved it on my own

[@connoribbotson1337]

Can someone explain to me this: when you graph the first equation, the big fat one, you get a different graph than when you graph the 8 / 2(1+x^2) (The 2 in the denominator is because it’s 2I). Does this just mean that those two graphs have the exact same area? Also, at what point in the video did the graph actually change, because I thought the starting equation was the same as the ending...

[@mihaiciorobitca5287]

sum( k=1 to n of Ak^-1) is (sum k=1 to n os Ak)^-1 where Ak €M2 Prove thet exist Ak with the element different of 0 Could somebody help me please ?

[@sam-kx3ty]

Please can you do this same way for a question like 1/x^4 +1 ? Pretty please!!!

[@Vivisions]

This was very satisfying...

[@andrewpod5693]

I have left with a little doubt. When we do substitution of integral variable(x to 1/x). Even if we call variable "x" in new integral isn't it actually different variable? And do we have rights to mix them in one equation?

[@runefullbody]

so where did the negative come from in the denominator

[@nicolascamargo8339]

Excelente

[@KrSaPoww]

Top tier

[@DevinSamarin]

I like big integrals and I cannot lie

[@strangemathematician1572]

What is the definite integration of the fractional part of secX from 0 to π/2

[@abedbob4046]

First of all no teacher in the world would give his students such integral, unless he hates all his students and wants them to fail. Second , even without exam, a teacher may give this kind of problems if for example come unprepared to teach ,keep the student quiet or get a day off.

[@sam-kx3ty]

I love you !!

[@leif1075]

Yes but QUESTION would anyone actually,think of this if they werent told it was true..I hope you can please answer..Thanks very much..

[@GooogleGoglee]

Answer to 3 decimale place: 6.283 ;-)

[@shrihari154]

One doubt at 4:16 since we have individually taken reciprocal for each function then we have to divide each function by x^2 as per the definite integral property???? hoping to get an answer. Thank you.

[@sugarfrosted2005]

#Kannada tech It's the chain rule. Also note that when we substituted in 1/x we flipped the integral as going from infinity to zero. Hope that helps. Try working it out with the u sub x=1/u

[@pasqualepaldino9811]

Can u adevice me a good book of calculus 2, prof. By the way you are the best☺

[@imagine_8681]

Wait how did yo get tan towards the end? I thought I’m done with trig lmaoo

[@JuanMataCFC]

amazing!

[@justinscheidler5938]

Would love to see geometric interpretations of these integrals. Might give some intuition? (Taking out of my ass probably)

[@jameswilson8270]

Coffin problem for sure! Very nice!

[@KnakuanaRka]

2:35 Wait, are there some qualifications on using that? Try a constant function like f(x)=1, and there’s clearly an issue.

[@hassanakhtar7874]

Its fine actually both integrals you would evaluate would diverge. Remember that when something diverges the equal sign really isnt meaningful.

[@glydon-w2w522]

This was brilliant 😍😍😍😍 But it was damn easy solved it by 30 sec ... By assuming x=tanT at first .. just took 5 steps

[@sarvasanjay7098]

Amazing

[@emmanuelalbazi8560]

Hello guys Can anyone please tell me how to solve this integral Integral from 0 to 1 e^xdx/sqrt(1-x^2)

[@MarcosGarcia-kk8lc]

That was beautiful.

[@RaspingBubbles6]

Can you do this using traditional methods (like intergration by parts ) cause I want to see how you get 2pi out of it

[@RaspingBubbles6]

Damn, I just wanted a really long video like (tanx)^1/3

[@blackpenredpen]

I integrated 1/(x^6+1) the traditional way tho. Check it out