
MHB Journeyman
#1
August 20th, 2019,
16:51
Show, that the identity
\[\int_{0}^{1}\frac{x^{n1}+x^{n\frac{1}{2}}2x^{2n1}}{1x}dx = 2\ln2\]
 holds for any natural number $n$.

August 20th, 2019 16:51
# ADS
Circuit advertisement

MHB Journeyman
#2
August 30th, 2019,
15:22
Thread Author
Here´s the suggested solution:

MHB Apprentice
#3
November 27th, 2019,
18:25
I'm sorry about poking a bit older threat, but I found this interesting, so here is my attempt:

MHB Journeyman
#4
November 30th, 2019,
14:25
Thread Author
Thankyou, Theia, for a clever solution!

MHB Master
#5
November 30th, 2019,
16:53
Originally Posted by
Theia
I'm sorry about poking a bit older threat, but I found this interesting, so here is my attempt:
Interesting. I note that Theia's solution uses a derivative of n, which implies that n doesn't have to be natural number! (I doublechecked this on WA.)
Dan

MHB Apprentice
#6
November 30th, 2019,
17:11
Yes, $ \displaystyle n \in \mathbb{N}$ is not a crucial requirement for $ \displaystyle F'(n) = 0$. But I haven't yet found a proof that shows what is required for $ \displaystyle n$.

MHB Apprentice
#7
February 1st, 2020,
19:16
Originally Posted by
Theia
I'm sorry about poking a bit older threat, but I found this interesting, so here is my attempt:
It really is an interesting question.
I've been thinking about it this evening, and wonder if what's below makes sense.