Find a sequence of real-valued nonnegative functions $f_n$ on $[0,1]$ such that lim sup is infinite, but limit of the integral is $0$

Context: I am looking for a sequence of functions $(f_n)$ on $[0,1]$ such that $\lim \sup_{n\to\infty}f_n(x)=\infty$ for all $x\in[0,1]$ and $$\lim_{n\to\infty}\int_0^1f_n(x)\text{d}x=0$$

Attempt: I'm looking at defining $f_n$ as a multiple of a characteristic functions, I am having a hard time meeting the conditions. I have looked at $f_n = n \chi_{[0,1/n]}$ and $f_n = n^2 \chi_{[0,1/n]}$. But none of those work.

Any help or guidance in defining this function is appreciated