Complex Analysis: Integral of ln(x)/(x^n+1) using Contour Integration

Today, we use complex analysis to evaluate the improper integral from 0 to infinity of ln(x)/(x^n+1) where our n is a real number greater than 2. We also come across some other familiar results which you can check out in more detail through these links to other videos I have created:

Solutions to z^n+1=0:
Integral of 1/(x^n+1):
Logarithm of complex numbers:
Complex exponential definition of sine:
Outtakes for integral over Gamma and gamma:

[mihaipuiu]

Your speech is perfect. The solution is the same. Thanks.

[FunctionalIntegral]

FIRST LIKE AND COMMENT.
You upload exactly want I want to see. Thanks a lot!
Keep up the great work and don't repeat what other math Youtubers do.
This is going to become an excellent channel!
Please continue with this topic. Complex analysis is certainly one of the most beautiful pieces of math.

[ozzyfromspace]

The integral you left as homework for us can be solved by considering e^(ix)/(x*n+1), and solving the integral the way you normally do. It works out without any complications. Then take the real part of the solution and, voila, you have your answer.

This video was posted months ago so it’s possible that I actually watched YOU solve that very integral months ago 😅. Thanks for making these awesome videos, I’ve really learned a lot from you.

And yeah, the result of the integral in your video was really spicy, I loved it 😭🙌🏽❤️💯🎊🔥

[wuyqrbt]

🤓👽🤓 ... That's it .
Great video (full of great points)
Thank you so much dear *QN³* ♥️

[eric]

Your Videos Are so Great, i really enjoyed your Video, And the Integral was really beautifull! Keep up the Great work😄

[wuyqrbt]

*Min: **25:00* .... *The Place Where the Magic happens!*
(I love this moment)

[vaibhavjain]

My brain has now been fried. Regardless, great work!

[isubhankar_gope]

I was trying to solve the problem using keyhole contour but did not get the right answer. Can you tell me if there is any rule to find a suitable contour for a problem?

[d.s.msahabandu]

For n=2, cot(pi/2)=0 so answer is wrong.

[edwardqian]

Why do you only take the first two poles, exp(ipi/n) and exp(i3pi/n)?

[ldego_]

You need to explore this posibility (28:19)

[davidemanuel]

Not sure if I missed it, but can somebody explain to me exactly why we can assume the principal branch for the logarithm?

[prakharrakhya]

What happens when you put n = pi in your final equation

[syedzaidi]

The integrand does not have a pole at z=0. If it did the integral (little gamma) would not vanish. Logarithm has a branch point at z=0. If you write out that integral, the "dangerous term is proportional to e x Log(e). Here e is the radius of that circle. That limit vanishes as e goes to zero. You have additional missteps and errors. The answer is not correct. Please check it out for n=3. The poles all lie on the unit circle and the angle of the return path (Psi) must be chosen carefully. It depends upon n. However it is still possible to give the result in closed form. As I said your answer is not correct.