[VA] Short & Sweet Math Challenge #25 "San Valentin's Special: Weird Math"

[Nihotte(lma)]

(02-14-2021 08:58 PM)Valentin Albillo Wrote:  Hi, all ! Happy San Valentín 2021 to all of you !


Concoction the Third: Weird Integral
[MRM: HP-15C and up]

Consider the following definite Albillo integral:



where Γ is the Gamma function, ln is the natural logarithm (i.e., base e) and φ is the Golden Ratio = (1+ √5)/2.


The Challenge:

Use your HP calc to compute (either manually or writing a program to do it) and output the value of the definite integral as accurately as possible, and then (the sleuthing part) use your HP calc (perhaps conduct some experiments) to try and attempt to answer this nagging question: What's so weird about this integral ?. (Again, forget about Googling for it because I concocted it myself and it's nowhere else to be found either.)

That's all. Enjoy ! ... and that's an order !

V.

Here is the program I used on my HP35s for the Concoction the Third and the Weird Integral:

LBL W
RCL P
x²
RCL X
-
LN
!
ENTER
ENTER // 2 times because ENTER has disabled moves in the stack...
RCL X
LN
!
+
/
RTN

With FN= W
1 and RCL P (where P is 1.61803398875)
∫FN d X
INTEGRATING gives 3.09016994375E-1 and it seems to be 1/(2*φ)