\[ \large I_n =\int_0^1 \frac{x^n}{x^2 + a^2} \, dx \]

Define the integral \(I_n\) as above for positive real variable \(a\) independent of \(x\) and natural number \(n\). Evaluate the limit below.

\[ \large \lim_{n\to\infty} \left [ \sqrt n \left( n I_n - \frac1{a^2+1} \right) \right ] \]

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