In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity: $$f(x)=x^2\exp{(−x^8\sin^2{x})}.$$ Intuitively, it can be expected that $f(x)$ is square-integrable as its peaks become more and more narrow. But how can one rigorously prove that $f(x)$ is indeed square-integrable?