1. Show, that the identity

$\int_{0}^{1}\frac{x^{n-1}+x^{n-\frac{1}{2}}-2x^{2n-1}}{1-x}dx = 2\ln2$

- holds for any natural number $n$.

2.

4. I'm sorry about poking a bit older threat, but I found this interesting, so here is my attempt:

Thankyou, Theia, for a clever solution!

6. 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 double-checked this on W|A.)

-Dan

7. 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 $\displaystyle n$.

8. 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.