TITLE: Evaluating a Lebesgue Integral
QUESTION [10 upvotes]: I have the following integral: 
$$\lim_{n \to \infty} \int_{0}^{1} \frac{n\sqrt{n}x}{1+n^2x^2} \, \mathrm{d}x
$$
To use the dominated convergence theorem I know that limit of $f_n$ is $0$ and $|f_n|<\frac{n^{1/2}}{2}$. However, I am having trouble to find a function that is greater than $\frac{n^{1/2}}{2}$ for all n. Can someone help? Thanks in advance.

REPLY [12 votes]: For fixed $x \in (0,1]$ the maximum of $\frac{n^{3/2}x}{1 + n^2x^2}$ is attained at $n = \frac{3^{1/2}}{x}$, providing an integrable dominating function:
$$\frac{n^{3/2}x}{1 + n^2x^2} \leqslant \frac{3^{3/4}}{4} \frac{1}{\sqrt{x}}$$