<s>[INST] Definite integral and limit $\lim_{n \rightarrow \infty} (n((n+1)I_{n}-\frac{\pi}{4})$ Given $I_{n} = \int_{0}^{1} x^{n} \arctan(x)dx $
Calculate:
$\lim_{n \rightarrow \infty} (n((n+1)I_{n}-\frac{\pi}{4})$
 [/INST] This is probably a too complex answer.
Considering $$J_n=\int x^n \tan^{-1}(x)\,dx=\frac{x^{n+1} \left((n+2) \tan ^{-1}(x)-x \,
   _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-x^2\right)\right)}{(n+1) (n+2)}$$ where appears the hypergeometric function, $$I_n=\int_0^1 x^n \tan^{-1}(x)\,dx=\frac{H_{\frac{n-2}{4}}-H_{\frac{n}{4}}+\pi }{4 (n+1)}$$ where appears the generalized harmonic numbers. So, $$A_n=n \left( (n+1)I_n-\frac{\pi }{4}\right)=\frac{1}{4} n \left(H_{\frac{n-2}{4}}-H_{\frac{n}{4}}\right)$$ Now, using the asymptotics of harmonic numbers $$A_n=-\frac{1}{2}+\frac{1}{2 n}+O\left(\frac{1}{n^3}\right)$$
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