Evaluate the integral $\lim_{n\to\infty}\int_0^1{\frac{nx\cdot \sin(x)}{1 + (nx)^a}}dx$

Question

How to evaluate the lebesgue integral of $$\lim_{n\to\infty}\int_0^1{\frac{nx\cdot \sin(x)}{1 + (nx)^a}}dx$$ for $a$ > 1. I tried solving this by $$|f_n(x)| \leq \frac{(nx)^{2-a}}{n}\frac{sin(x)}{(x)}$$ as sinx behaves similar to x near 0. But this leads me to believe it is not integrable .Help.

Answer 1

$\int_0^1{|\frac{nx\cdot \sin(x)}{1 + (nx)^a}}|dx=\frac 1 n|\int_0^n{\frac{y \sin(\frac y n)}{1 + y^a}}dy| \leq \frac 1n\int_0^n{\frac{y}{1 + y^a}}dy$. Clearly, $\frac 1n\int_0^1{\frac{y}{1 + y^a}}dy\to 0$. So consider $\frac 1n\int_1^n{\frac{y}{1 + y^a}}dy\leq \frac 1n\int_1^n{\frac{y}{y^a}}dy\to 0$ by direct evaluation.