A log trig integral with a beautiful result

Exactly what the title says...
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[@dhibaid3578]

Use properties of definite integrals, replace θ by (π/2 - θ) and add both I. Pretty straight forward and easier haha.

[@maths_505]

Oh yeah 😂 but this is more fun

[@user-lu6yg3vk9z]

@@maths_505why didn’t u use kings property?

[@famillemagnan1313]

Much easier indeed!

[@manjugangwar7245]

I used the same

[@orionspur]

Replacing theta with x, the curve is symmetric with respect to the segment from (0,2) to (pi/2,0), so the area of that triangle is pi/2.

[@nerden1549]

Love when he writes π as a и with - on top of it

[@vladthemagnificent9052]

ln, sin, and tan also looks like lu siu and tau

[@gabrielerizzoo]

i encountered this integral in school: integral from 0 to 1 of xe^x, i obviously solved it using integration by parts and it resulted in 1 but i tried to generalize it using a parameter alpha in the exponent of x so i got the integral of x^alpha*e^x but i'm not smart enough to solve it and i would love to see it in one of the next videos.
also i love watching your videos, especially at night since they're so relaxing and they help me falling asleep. great work! keep it up

[@rohitashwaKundu91]

Some people are somehow not happy with you not using that easy 2 liner solution. But (at least to me), this one was way more fun. I think sometimes, the way matters more than the result. And Ik it's better to have a simple solution, but still sometimes the complex one is the one which actually seems interesting and more fun. So thanks!

[@maths_505]

Thanks for getting it homie

[@SussySusan-lf6fk]

Yep 😊

[@CM63_France]

Hi,
"ok, cool" : 1:11 , 1:56 , 2:29 , 3:46 ,
"Terribly sorry about that" : 4:55 , 6:31 .

[@ericthegreat7805]

Basically this is saying the magnitude of sin2theta relative to tantheta along the first quadrant changes by a rate of pi/2 (up to a constant).

[@MrWael1970]

This is gorgeous, thank you very much.

[@LeoInPain]

Using the change of variable u=pi/2 - thêta, and addint it to the initial integral leads to the results much faster. (A bit frustrated for you not using it)

[@justdoingmath292]

The integral reduces to int(1)dx from 0 to pi/2 using King, Log, and Trig properties without substitution.

[@mikecaetano]

At the point where you expand by e^x/2 (9:08), the integrand readily becomes sech(x/2) when multiplied by 2. 1/2Integrate[Sech[x/2], x,-infy,infy] Nevertheless, the z-substitution afterward was nifty.

[@Anonymous-Indian..2003]

Ohh man just replace θ by (π/2 - θ)
And then add both.
You'll get,
2 I = ∫ { ln(tan²θ) / ln(tanθ) }dθ
= ∫ 2 dθ
= π
I = π/2

[@Haxislive766]

Here's the solution in 3-4 line --
I= ∫ [ (stuff) -1+1]d∅ now take lcm
With -1
U will find inside logarithms upstairs ln(sin2∅)
Downstairs ln(tab∅)
You have listen about The KINGS rule apply to the above integral and you will find the integral collapse 0 ,
∫d∅ from π/2to 0
= π/2 ans

[@mohnedhammadi4106]

The way that you use to solve the problem is so complex it’s easier to use king property and log property to get the results in 5 steps and by the way the result is pi/2not pi

[@kingzenoiii]

Greetings Kamal, i also forget how to talk from time to time 😂😂

[@giuseppemalaguti435]

Mi sembra troppo semplice 2I=Int(2)dx(0...π/2)=π..I=π/2

[@bandishrupnath3721]

i didnt understand the part where u say as x is an odd function but the others are even functions ,due to even f(x) the integral converges to zero from negative to positive infinity

[@lol1991]

If you think about geometry, at least in the case where the bounds are finite, since the interval is symmetric, the area on the right side of the origin is equal and opposite to that on the left side of the origin so they cancel

[@lol1991]

I mean the area is technically not negative but the value of the integral is the opposite

[@rageevanthaneswaran4659]

Is there any textbook for these kimd of question

[@3manthing]

It's all making sense now, the Black speech of Mordor, sounded very scary when I was a child, cause it was all maths.😁 I even got some hints for this to be true at school, when writting a test, and I looked to my neighbours test for a second, the teacher gave me a nasty look, and said "I see yooouuu."😅

[@leroyzack265]

A wild integral representation of pie.

[@HeyKevinYT]

Another dope integral representation of π

[@faaiz6672]

Which country are you from? Your are videos are so good

[@maths_505]

I'm from Pakistan

[@SuperSilver316]

No beta function??, are you okay??

[@trelosyiaellinika]

😂

[@davidblauyoutube]

Nice!

[@maths_505]

Thanks mate