Let $f(x)=\liminf_{n\to\infty} f_n(x)$ and let $g_n(x)=\inf_{k\ge n} f_k(x)$ so that we have $$ f(x) = \sup_n g_n(x). $$

As $g_n$ is an increasing sequence of measurable nonnegative functions we can apply the monotone convergence Theorem to obtain $$ \int_X f\, d\mu = \lim_{n\to\infty} \int_X g_n\, d\mu. $$ On the other hand, being $g_n\le f_n$ we conclude by observing $$ \lim_{n\to\infty} \int_X g_n\, d\mu = \liminf_{n\to\infty}\int_X g_n\, d\mu \le \liminf_{n\to\infty}\int_X f_n\, d\mu. $$