The trig integral of your dreams (or nightmares)
ฝัง
- เผยแพร่เมื่อ 5 ก.ย. 2024
- A fascinating trig integral with a surprising solution development and beautiful result.
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Advanced MathWear:
my-store-ef6c0...
OMG 8:33 the moment i realized how you used the tangent addition formula and that the integral would simplify I was fucking blown away. That is truly some anime type tricks right there by kamaal.
16 is 4^2 so in the final result you have something*((1/4)*Γ(1/4))^2 which simplifies to Γ(5/4)^2
Hi,
Nice formula : tan^-1 ( (1-z) / (1+z) ) = pi/4 - tan^-1 z . And I realized that I had already discovered this formula a few months ago.
"ok, cool" : 3:51 , 4:38 , 11:25 ,
"terribly sorry about that" : 9:26 , 9:40 , 12:44 , 13:04 .
In my msc in physics I derived a Hamiltonian for atom-light interactions for a quasi 1D system in terms of gamma(1/4), so the Lemniscate constant has "real world" applications
"The best way to simplify things is to make them more complicated." Yip!
I played with substitutions like
x = arctan(sin(theta))
u = 4x
v = Pi-u
w = Pi/2-v
y = 1/2w
and i have got result in terms of Elliptic integral
Pi/4*EllipticF(Pi/4,2)
This has been a wild integral, kamaal was playing 4D chess with that addition formula
When i started watching,it looked like a nightmare,but at the end this integral is adream!💯
Hey friend 😊. I did it just the same way. It was a nice one.
Solve this integral : (ln³x)/(x²+2x+2)
Limits being 0 to +♾️
Weierstrass has yet to disappoint me:)
arctan tan connection is the best thing I saw today
this is actually life-changing
Amazing substitution tricks!!
It is very interesting result and innovative solution plan. Thank you.
4:40. Wow! Talk about non-intuitive! Weierstrass sub hit that integrand like a cluster-bomb!
Harikasınız hocam
Jee advanced(india) exam 5 days left
From watching your integrals to getting a slap from chemistry,i came a long way
Impressive. I like the cut of your jib!
Also do try integral 0 to pi/2 of arctan(0.5sinx) dx
A much more interesting result
intgrl 0 to pi/2 arctan( a * sinx) dx. {at lim a tends to infinity} = pi^2 /4
I'm not joking.
As of your integral, it's pretty impossible as we have to solve
int 0 to 1/2 ln(a + sqrt(1+a^2)) /(a sqrt(1+a^2)) da, it's not possible by Feynman or anything.
But for my integral it becomes
int 0 to infinity ln(a + sqrt(1+a^2)) /(a sqrt(1+a^2)) da
This is easy if you use a=tanx and then split up the ln term. Finally after applying Feynman in one integral and applying geometric series in another, you get pi^2/4.
Hahahaha 1:36 . It happens when I practice Cal question
Hi, I hope you do vidéo about : int from 0 to 1 for (ln x)²/(x²+1) cuz its equals = pi³/16, i solved its by séries, btw Nice video
I solve with a x=1/t substitution, and combine to use the Residue Theorem.
What's the app name
Sooo I was playing on wolfram and found out that int ( 0 to infinite of x^ln(1/sqrt(x))) is exactly sqrt(2epi). You have any approach or reason for this?? Also written like that cuz is fun but better way is prolly x^((lnx)/-2)
It's not very hard
Substitute, lnx=t
int -inf to +inf, (e^t)^( - t/2) e^t dt
int - inf to +inf, e^( - t^2/2 +t)
int - inf to +inf, e^( 1/2 - ( t/sqrt2 - 1/sqrt2 )^2) dt
Take t/sqrt2 - 1/sqrt2 =u
1/sqrt2 dt = du, dt=sqrt2 du
int -inf to +inf, e^1/2 e^(-u^2) sqrt 2 du
Apply gaussian integral result
sqrt(pi) * sqrt(e) * sqrt(2)
sqrt(2epi)
Con lo sviluppo in serie della arctg e la beta function trigonometrica risulta I=(1/2)Σ((-1)^k/(2k+1))β(k+3/4,1/2)..poi..???
What do you mean when you said (third world ciuntry)? since you are solving a scientific math problem, probably some people not like these kind of expression to be used while you are talking math...
My dear
Ok Cool !
Hey bro (I'm who was calling you master)
I have one new challenge for you!
Int 0 to 1 [ ln ( 1-x² ) ] dx
This is my today's mock test question 🌟
Factorise the argument of the ln function and then apply log properties. You'll get the sum of 2 integrals. Then go for integration by parts.
OmG 😂
But they've used ln expansions and some ultimate series simplification in given solution
You're legend bro
(Master )
@@sarahakkak408 indeed it doesn't 😂
He could have meant floor function by [ ]
noice