Function $f(x)$ is continuous on the interval $[0,1]$. Then find the value of the limit $\lim _{n \rightarrow \infty} \int_0^1 \frac{n\left(x^2 \sin ^2 x+\cos x\right)}{1+n^2 x^2} d x$
a $\pi$
b $\frac{\pi}{6}$

C $\frac{\pi}{3}$
d $\frac{\pi}{2}$

only answer

Function $f(x)$ is continuous on the interval $[0,1]$. Then find the value of the limit $\lim _{n \rightarrow \infty} \int_0^1 \frac{n\left(x^2 \sin ^2 x+\cos x\right)}{1+n^2 x^2} d x$ a $\pi$ b $\frac{\pi}{6}$ c $\frac{\pi}{3}$ d $\frac{\pi}{2}$ only answer

This problem explores the limit of an integral involving a continuous function on [0,1] with terms containing $x^2 \sin^2 x$ and $\cos x$. Using advanced calculus techniques, including the Dominated Convergence Theorem and the Fundamental Theorem of Calculus, the solution yields $\frac{\pi}{2}$. Suitable for undergraduate mathematics students.

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