Integral of sequence of functions

Question

Let be $f_n(x)=\frac{nx}{1+nx^3}$ a sequence of functions

Prove:

$$\lim_{n\to\infty} \int_{0}^{1}f_n(x) dx =+\infty$$

Answer 1

Note that $$\int_0^1\frac{nx}{1+nx^3}\mathrm{d}x\gt\int_0^1\frac{nx^2}{1+nx^3}\mathrm{d}x=\left[\frac13\ln{\left|1+nx^3\right|}\right]_0^1=\frac13\ln{(1+n)}\to\infty$$

Answer 2

Or $$\int_{0}^{1}\frac{nx\,dx}{1+nx^3}\geq \int_{0}^{1}\frac{nx\,dx}{1+nx^2}=\frac{1}{2}\log(n+1). $$ The actual divergence speed is $\sim\frac{2\pi}{3\sqrt{3}}\sqrt[3]{n}$.