# Limit of Definite Integral #1

Calculus Level 5

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

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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