To show that $\exp(\psi(2n+1))\int_0^1x^n(1-x)^n dx$ is a positive integer, where $\psi(x)$ is the Chebyshev function

Question

I have to show that $$\exp(\psi(2n+1))\int_0^1x^n(1-x)^n dx \ \in \mathbb{N}$$ where $\psi(x)$ is the Chebyshev function, and $0 \leq x \leq 1$

The Chebyshev function is defined as the following: $$\psi(x)=\sum_{p^{\alpha}\leq x}\log p$$

For instance, when $n=3$ we have $$\exp(\psi(7)) = \prod_{p^{\alpha}\leq x}p=2\cdot3\cdot 2\cdot5\cdot7$$ $$\int_0^1 x^3 (1-x)^3 \ dx = \frac{1}{140}$$ $$2\cdot3\cdot 2\cdot5\cdot7\cdot\frac{1}{140} =3$$

Answer 1

If you have a copy of Ram Murty's $\textit{Problems in Analytic Number Theory}$, 2nd Edition, it's exercise 3.1.4.

Show that $e^{\psi(n)}=\text{lcm}\{1,2,\cdots,n\}$.

Show that $I_n=\int_0^1x^n(1-x)^n\ dx\leq 2^{-2n}$.

Show that $I_n=\sum_{k=0}^{n}\int_0^1(-1)^k{n\choose k}x^{n+k}\ dx$ and that $I_n\times\text{lcm}\{1,2,\cdots,2n+1\}$ is a positive integer.

Think you can take it from there?