The trig integral of your dreams (or nightmares)

A fascinating trig integral with a surprising solution development and beautiful result.
My complex analysis lectures:
• Complex Analysis Lectures
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[@user-dm1tm8uw2o]

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.

[@Jocularious]

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

[@nizogos]

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

[@CM63_France]

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 .

[@worldnotworld]

"The best way to simplify things is to make them more complicated." Yip!

[@stefanalecu9532]

This has been a wild integral, kamaal was playing 4D chess with that addition formula

[@yoav613]

When i started watching,it looked like a nightmare,but at the end this integral is adream!💯

[@Mathematician6124]

Hey friend 😊. I did it just the same way. It was a nice one.

[@holyshit922]

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)

[@Jalina69]

arctan tan connection is the best thing I saw today

[@user-gj2kw7wx2q]

Weierstrass has yet to disappoint me:)

[@tzovgo]

this is actually life-changing

[@neg2sode]

Amazing substitution tricks!!

[@MrWael1970]

It is very interesting result and innovative solution plan. Thank you.

[@shubhammeghani8232]

Solve this integral : (ln³x)/(x²+2x+2)
Limits being 0 to +♾️

[@xxxx015]

Harikasınız hocam

[@KramRemin]

4:40. Wow! Talk about non-intuitive! Weierstrass sub hit that integrand like a cluster-bomb!

[@Ghostwriter_zone]

Jee advanced(india) exam 5 days left
From watching your integrals to getting a slap from chemistry,i came a long way

[@dthez4768]

Impressive. I like the cut of your jib!

[@sidhantmohanty5256]

Also do try integral 0 to pi/2 of arctan(0.5sinx) dx

[@SussySusan-lf6fk]

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.

[@shamgermedad9560]

Hahahaha 1:36 . It happens when I practice Cal question

[@MugdhoDas9]

What's the app name

[@toufikakkak8459]

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

[@vascomanteigas9433]

I solve with a x=1/t substitution, and combine to use the Residue Theorem.

[@Anonymous-Indian..2003]

Ok Cool !

[@omkarjoshi9137]

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)

[@SussySusan-lf6fk]

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)

[@giuseppemalaguti435]

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..???

[@Akhulud]

noice

[@joelchristophr3741]

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 🌟

[@maths_505]

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.

[@joelchristophr3741]

OmG 😂
But they've used ln expansions and some ultimate series simplification in given solution
You're legend bro
(Master )

[@maths_505]

@@sarahakkak408 indeed it doesn't 😂

[@SussySusan-lf6fk]

He could have meant floor function by [ ]