$I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n}dx$ is increasing and goes to $\infty$

Question

Let $a$ and $b$ be real numbers with $a< b$, and let $f$ and $g$ be continuous functions from $[a, b]$ to $(0, \infty)$ such that $\int_a^b f(x) dx = \int_a^b g(x) dx$, but $f \neq g$. For every positive integer $n$, define \begin{equation} I_n = \int_a^b \frac{(f(x))^{n+1}}{(g(x))^n}dx \end{equation}

Show that $I_1, I_2, I_3,...$ is an increasing sequence with $\lim_{n \rightarrow \infty} I_n = \infty$.

I would like a hint for this problem.