A wild quintic trigonometric integral

[@manstuckinabox3679]

- So, our methods are pretty much the same, but I used a generalized form of the Mellin transform of 1/1+x^b (proved using contour integration, review Gamelin's exercise on it) at 1 and at 3 to get 2*(pi/5sin(pi/5) + pi/5sin(3*pi/5)) ... - hey, I thought of beta too! it's amazing how contour integration and beta function are so similar when it comes to solutions, I think Churchill had a nice exercise exploring the beta function in light of complex analysis. Man, I'm loving the content my man! keep it up! and don't fear experimenting with the content as well!

[@MatthisDayer]

i see 5, so the answer must be something with golden ratio

[@vascomanteigas9433]

Using the substitution x = arctan(t), which turns the integral domain to the entire positive real axis, and simplifying the rational function gets: (t^2+1)/(t^4-t^3+t^2-t+1) Which is an even function. Applying the residue Theorem solves the problem, but all four poles (only two are enclosed by the contour) are irrational.

[@jieyuenlee1758]

11:32 asumme y is the golden ratio Sqrt5=2y-1 ysqrt5=2y²-y ysqrt5=2y+2-y ysqrt5=y+2 the denominator can be simplyfy to 5sqrt(y+2) where y is the golden ratio.

[@orionspur]

Interesting. I probably would have tried factoring the denominator, and simplifying.

[@MrWael1970]

Very interesting integral, and very smart solution plan. Thank you.

[@TheArtOfBeingANerd]

*radical sign and five* Coorporate needs you to find the difference between these two images. Maths 505: they're the same picture

[@giuseppemalaguti435]

I=(π/√5)(1/√(5-2√5)+1/√(5+2√5))+(2/√5√(5-2√5))arctg1/(√5-2√5)-2/(√5√(5+2√5))artg(1/√(5+2√5))=3,46...

[@jwkim4428]

How about this form, (2pi/5 sec pi/10)*phi^2 😅

[@The_Shrike]

Just got your gamma function hoodie, will be wearing it to school on Monday

[@maths_505]

WOAH 🔥

[@jurgensand9201]

The golden ratio satisfies fi*fi = 1-fi and not fi*fi = 1+fi. With this, the result is slightly different but similarily compact 😊

[@Marvels18]

Nicee

[@Haxislive766]

Just not hold on ß function yet still the same procedure I apply earlier

[@Marvels18]

Nicee