Math for fun: integral of ln(x) from 1 to ? is equal to 2 

Get started with a 30-day free trial on Brilliant: 👉brilliant.org/blackpenredpen/ ( 20% off with this link!)

In Finding Nemo they could have just used the Lambert-W function and then they would easily find Nemo

Yay the fish is back

the fish function is kinda bad imo, you give away a medium fish and a small fish for a big fish but I personally find that since you have to eat the fish in one go or it'll go bad the m fish and s fish serve a better purpose for consumption and if you are collecting fish, I doubt you want to decrease the number of fish you have unless your tank is running out of space, but bigger fish generally need more space to be healthy so I wouldn't use this as a solution even then

Fun fact, the general form with an area of N under the curve is t = (N-1)/W((N-1)/e)

that's what I thought right away when I saw the question :)

Wow first real practical application of the Lambert W function I've ever seen that could come up in everyday calculus! Thanks so much!!

What an interesting problem! I could see things like this being used for many applications, like how long you need to drive to drive a certain displacement.

Hey bprp, I just wanna say a massive thank you for your videos, not only are they super entertaining for me but when I started watching them a couple years ago I had no idea what was going on, not about calculus, complex numbers or anything. But slowly I started picking things up and I started to basically get a online class from you, whilst being entertained with great content. Now that my curriculum is more lined up with your content and european style (IB by the way), I actually use the stuff you use a lot and it helps me get really good grades, so yeah sorry for the paragraph but thanks :)

Thanks to your videos I was actually able to solve this one on my own, never thought I would like math this much. Thank you for reigniting my interest in math :)

absolutely yes I love todays lesson and i began to understand more equations like this thanks for sharing this video and yh i tried to do this on my own too but thanks again for teaching me this

Nice question and good answer, love it.❤

I love the proof by wolframalpha double check at the end.

Paused the video, tried it first, comes out approx 3.591. 1 / (Fish 1/e) Edit : Fixed typo

It's crazy how after watching so many of your videos, I can solve problems like these without any problem at all in a matter of seconds. Lambert W fuction really is so good afterall.

Such a common problem man, thank you for the solution.

Evaluate the surface integral ∬SF⋅dS for the vector field F(x,y,z)=x^2i+y^3j−xzk, where S is the part of the paraboloid 0.6+1.1x^2+y^2−z=0 that lies above the disk x^2+y^2≤1 and has upward orientation

Can you do a clear video of the Lambert function for high school students? Thank you.

Gosto muito da condução dos seus vídeos, da forma como faz matematica.

t can be further simplified to just 1/W(1/e). I still have not taken full calculus classes but I do love the lamber W function

The result can be simplified to 1/W(1/e)

For anyone wondering you can simplify the result like this: = e^(W(1/e) + 1) = e * e^W(1/e) = e * W(1/e) e^W(1/e) / W(1/e) = 1 / W(1/e)

You can also use parametrical equations to integrate ln x y=t x=exp(t) $ydx dx=exp(t)*dt $t*exp(t)dt You don’t get out of integrating by parts but it’s a little more intuitive.

please try this equation: e^i^x=x

Instead of using the lambert function and then solving numerically, you can just use recursion to solve numerically. Since t.lnt-t=1 then t=(1+t)/ln(t). Let ans equal some starting number e.g. 10 and then ans=(1+ans)/ln(ans) and repeat until it converges e.g. 4.777239, 3.694211, 3.592233,] 3.591122, 3.591121. It converges pretty quickly and easy to do on a calculator.

Please make video on Fourier transform.

Fun fact, I actually did this question and t can also be write as the 1/LamberW(1/e), i feel like the method you used was harder than mine though but its fun to see that e^( LampbertW(1/e) + 1) is the same as 1/LamberW(1/e) seems like it wouldn't be true😀

plz make a video and explain a simple solution for Morley's theorem.🙏

i dont know about W function cuz i am in high school . i tried and got t should be between 3 and 4

I guarantee the answer uses Lamdba W function!!

Integration of lnx, by parts: u = lnx, u' = 1/x v' = 1, v = x 2 = [uv - integration of u'v] 2 = [xlnx - integration of 1] 2 = [xlnx - x] Upper bound is t, lower bound is 1 2 = tlnt - t - ln1 + 1 2 = tlnt - t + 1 1 = tlnt - t 1 = t(lnt - 1) Iiii have no idea how to simplify and/or solve this

I knew the answer was going to use Lambert W function the second I saw the problem)

My solution was pretty similar to that in the video, but I put the answer in a neater form (as some others have in their comments) ∫₁ˣln t dt=[t(ln t-1)]₁ˣ (using integration by parts) =x(ln x-1)-1(0-1) =x(ln x-1)+1 So we need x(ln x-1)+1=2 x(ln x-1)=1 x(ln x-ln e)=1 x ln(x/e)=1 (x/e)ln(x/e)=1/e ln(x/e) e^[ln(x/e)]=1/e So (as 1/e>0) there is a unique real solution, which is ln(x/e)=W₀(1/e) x/e=e^W₀(1/e) [⇒x=e^[W₀(1/e)+1] ] But W₀(a)e^W₀(a)=a, so e^W₀(a)=a/W₀(a) So x/e=(1/e)/W₀(1/e) x=1/W₀(1/e).

Could you use series expansion of ln(x) instead of W?

Nice!

Bro plz try this d/dx(log base x of n)

The W function is my favorite function I never learned about in a proper math class!

the lambert w function is bprp's favorite math concept

hey bprp, could you try making a vid about tetralog someday? I wld appreciate if you do.

wait it said that you have to pay for math and science subject is their a free trial for it or is it have to be paid

Dam I saw it from the thumbnail and thought we got to choose the lower bound too and had an answer since we could freely choose one Not my first Lamberto tho. I see ln, I know. Get integral and factor out a b to be nicer. b(lnb-1) = 1 Now set up lambo by creating an e^ term. Then have it match what’s next to it by multiplying e^-1. e^(lnb) (lnb-1) = 1 e^(lnb-1) (lnb-1) = e^-1 Lambo: lnb-1 = W(e^-1) Finish: b = e^[W(e^-1)+1]

Integral entre 0and1(-ln(x))^1/2

Try this insanely hard integral : Integral between 0 and pi/2 of tan(x)^(1/(x^x))

hey bro... Where are you from?? I'm from Bangladesh... Your videos are so helpful for me... Some days later my year final examination... I'm derepressed 😭😭.... Take my love ❤

Is there any way to expand ( a+b+c+d)^7

Whenever I see problems that use the Lambert's W Function, I just say, nah too lazy, and end up using the Newton-Raphson Method 😂😂

Please make a detailed video on fields medal

Well did it after expanding logt upto t-1 or even upto of power 3 from there we can get an approx value 1+ (2)^1/2 instead using lambert transformation

Your help please Integral of ((x^2 + sin x)/(x^2 +1))dx

I got the solution, but I had trouble checking it.

Если использовать формулы для функции Ламберта, то ответ можно упростить: t=e^(1+W(1/e))=e*e^W(1/e)=e*(1/e)*1/W(1/e)=1/W(1/e) Получаем: t=1/W(1/e) 😜

Hey Steve! Show us a HARD analysis problem - at the 899 (end of PhD or even postdoc) level. 😈

Lambert W function ❎ bprp's fish function ✅ 😊

X = e^(W(3/e) + 1)

I've even asked WolframAlpha this question, but it seems to bug out and ask me to reask. When I formulate it another way, it just shows me a numeral approximation but nothing else. Could you try and solve it, please? Thank you! x^x^x = 2

Though I know how to solve this I came to see because to see bprp's Lambert fish function. 😅

Lambert what?

You can put the answer in an alternate form as follows: t=1/W(1/e), which looks better in my opinion.

What if I want that area to be i?

Just checking in to see how you pronounce ln x.

can you please come to Iran someday? pleaaaaaaaaase

W still feels like cheating. Or "here magic happens".

I have to say, I don't remember what the antiderivative of ln(x) was...I suppose if it was that straightforward, lol, there wouldn't be a video about this...maybe it doesn't have a straightforward derivative?...I don't know, lol...

Can you prove that why (-)(+)=- (-)(-)=+

I don't think I've actually understood what is the W function

Please Make a video with a youngest an and famous mathematician and physician subbrno isaac bari

Try this very difficult integral : ∫_0 ^(pi/2) tan(x)^(1/x) dx. So you want a VERY difficult question ? Solve the differential equation : ∫_0 ^pi ( ∫_y' ^y'' ) dx dt is y'''(e).

B = (84 - A)/(1 + A) (A and B both are positive integers. A, B > 1 ) then find the value of AB.

i tried it; my brain was hurting i was so angry but i got 3.5911223...

I am the first one commenting on your video

I dont understand the W function, can someone explain? It doesnt seem to give any information at all. Why not just invent some function K so that t = K(2)?

The first one to comment on! 🕶

I am the 300th viewer

I think i'm first ?

first

Stop making easy exercices, I feel that I am good at math

Jesus loves you ❤️ Please repent and turn to Him and receive Salvation before it's too late. The end times written about in the Bible are already happening in the world. Jesus is the son of God and He died for our sins on the cross and God raised Him from the dead on the third day. Jesus is waiting for you with open arms but time is running out. Please repent and turn to Him before it is too late. Accept Jesus into your heart and invite Him to be Lord and Saviour of your life and confess and believe that Jesus is Lord, that He died for your sins on the cross and that God raised Him from the dead. Confess that you are a sinner in need of God's Grace and ask God to forgive you for all your sins through Jesus. Jesus loves you. Nothing can compare to how He loves you. When He hung on that cross, He thought of you. As they tore open His back, He thought of your prayer time with Him. As the thorns dug into His head, He thought of you spending time reading the Bible. As the spears went into His side, He imagined embracing you in heaven. Please repent and turn to Jesus now before it's too late. He is waiting for you with open arms but time is running out.

Нашёл t≈3.5911215, решал через функцию Ламберта W(x).