Bernoulli number type asymptotics

I find an interesting formula but I can not prove it. Show that
$$I_n=(-1)^{n+1}\int_0^1 B_{2n+1}(x)\cot(\pi x) \, dx\sim\frac{2(2n+1)!}{(2\pi)^{2n+1}}$$
where $B_n(x)$ is the Bernoulli Polynomials.

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According to Corollary 1 of [1],

(-1)^{n+1} \frac{(2\pi)^{2n+1}}{2(2n+1)!} B_{2n+1}(x) \longrightarrow \sin(2\pi x)

uniformly on compact subsets of $\mathbb{C}$. It follows that

(-1)^{n+1} \frac{(2\pi)^{2n+1}}{2(2n+1)!} \int_0^1 B_{2n+1}(x) \cot(\pi x)\,dx &\to
\int_0^1 \sin(2\pi x)\cot(\pi x)\,dx \\
&= 2\int_0^1 \sin(\pi x)\cos(\pi x)\cot(\pi x)\,dx \\
&= 1,

and so

(-1)^{n+1} \int_0^1 B_{2n+1}(x) \cot(\pi x)\,dx \sim \frac{2(2n+1)!}{(2\pi)^{2n+1}}

as $n \to \infty$.

[1] Dilcher, K. Asymptotic behaviour of Bernoulli, Euler, and generalized Bernoulli polynomials. (ScienceDirect link)