Find the limit
$$\lim_{n\to\infty}\int\limits_{0}^{1}{\frac{n\cos^2x}{1 + n^2x^2}\mathrm{d}x}$$
I know I should start by subsituting in $y = nx$, but I don't know where to go from there
2026-05-16 00:51:03.1778892663
Find the limit of this integral $\int\limits_{0}^{1}{\frac{n\cos^2x}{1 + n^2x^2}\,dx}$ as $n \to \infty$.
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Hint. By the change of variable $y=nx$ one gets, as $n \to \infty$, $$ \int\limits_{0}^{1}\frac{n\cos^2x}{1 + n^2x^2}\:dx=\int\limits_{0}^{n}\frac{\cos^2\frac{y}{n}}{1 + y^2}\:dy=\int\limits_{0}^{\infty}1_{[0,n]}(y) \cdot\frac{\cos^2\frac{y}{n}}{1 + y^2}\:dy \to \int\limits_{0}^{\infty}\frac{dy}{1 + y^2}=\frac \pi2 $$ where one may used the Dominated Convergence Theorem.