Complex Analysis: Integral of log(sin(x)) #2

Today, we re-evaluate the integral of log(sin(x)) using another complex analysis approach. Below is the first approach I used a while back.

[TheHellBoy]

i just realised i wasnt subscribed. you earned a sub for the most detailed videos on complex analysis

[ritwinnarra]

Really nice method! In this case, the part about checking arguments while splitting the logs isn't necessary, because we know the integral will be real, so during the evaluation we may disregard any purely-imaginary part, and then take the real part of the final answer.

[abdulllllahhh]

man you don’t know how much I love your videos, I’m in my senior year of high school and I’ve honestly spent 100s of hours watching your vids and trying these questions on my own. You’re channel is probably the best place to learn complex analysis from, or at least the best place to apply it and solve some questions. Thank you so much and keep doing what you’re doing

[RashmiRay-cy]

How do you balance the arguments if you pull e^ix out of the log rather than e^(-ix)? The argument of ln(e^ix - e^(-ix)) is pi/2 . ln(e^ix - e^(-ix)) = -ix + ln (e^2ix - 1) + 2pi i k. Setting x = pi/2 in the log gives ln (-2) whose argument is pi. The argument of -ix is 3pi/2 and thus 3pi/2 + pi + 2pi k = pi/2 > k = -1.Is this correct? How do we see the cancellation of the imaginary terms arising from the integrals of -ix and -ln i?

[khiemngo]

Nice solution! However, video at 13:57, how do you know that the singularity z = 0 is removable ? I think residue theorem could have been used here instead of the Cauchy Integral Formula.

[nathanpannifer]

Any chance you you do the integral of a cauchy distribution multiplied by a gaussian (both with different centers and widths)?

[thushariamarasinha]

Just Amazing...but can you make video on contour transformations when there's a substitution for z

[Playingcivilization]

Hi
Could you please do a video about integral from 0 to infty of ln(x)/((x+1)*x^{3/4}). I was trying to solve it using complex integration but failed to find proper contour:(

[Samiul_]

Would you make a video with Eddie Woo?
Or BlackPenRedPen

[dihinamarasinghe]

Yo can you help me solve this savage integral ( integral from 0 to 1 of (2-x)^2 × sqrt (x/(1-x)) dx ) by the method of complex analysis I keep getting the negative value of the actual value which is 11pi/16

[RozarSmacco]

Youre very smart yet seem to have missed entire point of Lavalier Microphone joke? am I missing something attempt at humor? You’re funny as Michael penn

[nurhaudi]

Sir, I have a nightmare problem that has not been resolved until now, here is the problem :

[0→∞] ∫ e⁻ˣ.∛(sin(x) + x) dx

Help me please!!!