Integral ln(sin x) from 0 to pi/2

In this video, yet another Peyam Classic, I calculate the integral of ln(sin x) from 0 to pi/2 using a clever u-substitution. Thank you Zach Lee for giving me the main ideas for the proof!

[AndDiracisHisProphet]

you know, you didn't change a bit in the last 11 years :D

[tomking]

2:27 I thought cosine was named after this property, not the property was discovered of cosine (cosine = sine of COmplimentary angle).

[srpenguinbr]

Wow. I remember watching the solution for this integral a long time ago. I didn't understand a single word, not ever u subs. Now I did it by myself. Thank you, math YouTubers.

[KidNamedVashin]

I wish I was as excited about life as he is about math

[reinerwilhelms-tricarico]

I tried that integral and got nowhere in circles. Gladly I gave up soon enough and found this. In fact, I searched for the expression “ln(sin(x)) dx” on YouTube and this was at the top of the list. Fantastic - I wouldn’t have guessed this way out of the treadmill.

[PackSciences]

It's also equal to the integral from to pi/2 of x*cot(x) dx.
( integration by part, and ln(sin(x))' = cot(x), and x' = 1)
It makes the exercise slightly more complicated because you need the two tools of integration: Integration by Part and U-substitution.

[helloitsme]

11:00 if you use the subtitution v=π-u it's easier cause sin(π-u)=sin(u) and the bounds become π/2 and 0 and the minus will flip the thing

[tgx]

My initial idea was the same, but when I got integral(ln sin(y)) for y in (0;pí), I did not see the continuation. Thank you very much.

[planet]

You can directly prove 4:12 by complimentary property,
Thanx for solution

[skillogify]

That is certainly an intro that i have seen

[ashwinvishwakarma]

I did it in almost the same way, except I immediately used symmetry argument when I got 1/2Int[0, pi] ln(sin(u))du=Int[0, pi/2]ln(sin(u))du.

[Anders]

You are insane!! Great video and great solution for the badass integral! I like your energy in these videos, professor!

[MrHansi]

it's a magic, i just prepared for my lesson and i found this magic decision . Thank you

[Antwelfare]

When I got to the part with
int[0, pi] ln(sin(x))dx I used the fact that
sin(x)=sin(pi-x).
It’s cool how you can solve this integral without actually integrating! :)

[MrQwefty]

All you needed to do was find the x, Zach Lee.... Exactly ;)

[cycklist]

Can we have the link to Zach's channel please?

[thomasblackwell]

And he asks what he is doing. I am so lost I could not find downtown Albuquerque even if I was sitting there at a traffic light with a flashing sign saying “You are in downtown Albuquerque.”.

[varunviswanath]

Why didn't you just use integration by parts taking ln(sin x) as u and 1 as v?

[xcalibur]

The area is negative...hmmmm.. so does that mean the function takes up only negative values between the interval 0 to π/2 ?

[MrRyanroberson]

Alright. Let's see which magic proof allows this to be non (-infinity)