Another approach is to multiple the ln(cos) by two (later divide everything by 2) which gives ln(cos^2), then define the integral with the Feynman approach in the natural log: ln( 1+ a(cos^2 -1))= ln(1-a sin^2). Then take the derivative with respect to (alpha=a) and then solve -with all the sin^2 terms in the integral - allowing factoring.
I use a TI n-spire CX CAS to solve these tough integrals. Usually, I have to manipulate them using Feynman’s technique, but differentiating with respect to the parameter is automatically done in the integral. Then integrate the result with respect to the parameter as soon as there is no integral sign and add c. Solve for c, using a specific value of the parameter (usually 0), and finally, evaluate the I(a) using the non-integral representation and the desired a-value.
By force of will! You thereby define a function I(alpha), which is, by definition, whatever the value of the integral is for any given alpha. You then differentiate with respect to alpha, and you can then figure out I(alpha), as the video shows. Then plug in alpha = 1 and you can get the value of the original integral.
Sir , I am at my initial stage of learning calculus I had completed my elementary and basics of calculus and now I am entering into college I want to learn more about calculus specially integrals plz tell me from where to start learning it 🙏🏽
Just keep solving integrals....you can find them all over TH-cam and maths stackexchange.....the books mentioned are also good choices....along with the one I plan to write eventually 😂
Wow! What an almost coincidence. In the last video I commented an exact similar integral just with x² instead of sin²(x). Did you get the inspiration or it's just a pure coincidence?!
Great video. I like such integrals which yield real answer instead of digamma,eta function and other special function based answers. I mean if you have to use wolfram alpha for finding values of digamma function, why not ask the integral itself to it. Anyways it is my take. Others might think differently.
watched this at 1.30 am and I have never been more sure that I want to keep studying maths. so cool
Hi,
"ok, cool" : 2:35 , 3:30 , 4:38 , 5:28 ,
"terribly sorry about that" : 8:02 , 9:58 , 15:44 .
Fantastic solution. Expressing cos and sin in terms of tang is key to solving this fine integral
Super cool solution, very two-rootful
it was wild and beautiful. it was equally root 2 rooted. This is an amazing video to conclude this wonderful day.
So early the thumbnail hasnt even loaded yet
same
Another approach is to multiple the ln(cos) by two (later divide everything by 2) which gives ln(cos^2), then define the integral with the Feynman approach in the natural log: ln( 1+ a(cos^2 -1))= ln(1-a sin^2). Then take the derivative with respect to (alpha=a) and then solve -with all the sin^2 terms in the integral - allowing factoring.
That was tooooo beautiful solution btw nice integral
I use a TI n-spire CX CAS to solve these tough integrals. Usually, I have to manipulate them using Feynman’s technique, but differentiating with respect to the parameter is automatically done in the integral. Then integrate the result with respect to the parameter as soon as there is no integral sign and add c. Solve for c, using a specific value of the parameter (usually 0), and finally, evaluate the I(a) using the non-integral representation and the desired a-value.
Thank you for fruitful effort.
the fraction at the end can be derarionalized into
2 ln ( sqrt(2) + sqrt(alpha) )
The root of all math videos
Awesome Kamaal! Too (2) rootful indeed! 😄
❤❤❤
Excelente el video
How can you introduce a perimeter alpha ?
By force of will! You thereby define a function I(alpha), which is, by definition, whatever the value of the integral is for any given alpha. You then differentiate with respect to alpha, and you can then figure out I(alpha), as the video shows. Then plug in alpha = 1 and you can get the value of the original integral.
@@worldnotworld by the force of will is seriously the best explanation I've ever heard 😂😂
Perimeter 😭😭
There is a channel called "The Feynman technique" that has hundreds of videos demonstrating this technique.
Very rootiful!
why not beta function
Sir , I am at my initial stage of learning calculus I had completed my elementary and basics of calculus and now I am entering into college I want to learn more about calculus specially integrals plz tell me from where to start learning it 🙏🏽
i have heard inside interesting integrals is a good book. Not sure about what level it is though
I recommend " a treatise on the integral calculus" Joseph Edwards, start from Vol I
Just keep solving integrals....you can find them all over TH-cam and maths stackexchange.....the books mentioned are also good choices....along with the one I plan to write eventually 😂
Un idea può essere cosx=√(1+cos2x)/2..sinx=√(1-cos2x)/2
I've a qustion, when we integrate like ( cos(lnx) )we can subtitue cos(lnx) by the real part of x^i then integrate, what's the proof of that?
Comes from Euler‘s formula. e^ix = cosx + isinx. Just let x = lnx
@@SnowboardAddict37
I know, but what is the proof of thag we can get out the ( Re)
@@jejnsndn It's the real part since e^ix = cosx + isinx that means that Re(e^ix) = Re(cosx +isinx) = cosx. Since the imaginary component is sinx
Wow! What an almost coincidence. In the last video I commented an exact similar integral just with x² instead of sin²(x). Did you get the inspiration or it's just a pure coincidence?!
Yes there was inspiration from that. Thanks mate.
You're welcome
Did you see my solution on that comment?
@@SussySusan-lf6fk yes i just need some time because i dont know Fourier transform yet
what app do you use to do math?
Ok, cool!!
Great video. I like such integrals which yield real answer instead of digamma,eta function and other special function based answers. I mean if you have to use wolfram alpha for finding values of digamma function, why not ask the integral itself to it. Anyways it is my take. Others might think differently.
The reason why, the reason why.
The reason why I had to die.
Did I bleed the blood of greed,
What was my destiny?
Dumb way to solve why we just put alpha tf