When $\int_a^b\left(f(x)\right)^kdx=\left[\int_a^bf(x)dx\right]^k$

Question

Under what conditions on $f(x)$ the following equation holds? $$\int_a^b\left(f(x)\right)^kdx=\left[\int_a^bf(x)dx\right]^k$$ with $k\in\mathbb{N}$ and $k\gt1$. I know the following inequality holds: $$\int_a^b\left(f(x)\right)^kdx\ge\left[\int_a^bf(x)dx\right]^{k-1}$$ when: $$\int_a^bf(x)dx\ge(b-a)^{k-1}$$ but is it possible from this last two inequalities say something about the first one? Thanks.