If $f(x)$ is integrable can we say that $f(x)^n$ is integrable?

Suppose $f(x)$ is a positive and continuously differentiable functions. In addition, it is well-known that $\int_{0}^{\infty} f(x)dx$ is bounded. My point of view is that $\int_{0}^{\infty} f(x)^mdx$ (where $m \in \mathbb N$) is bounded. I will be gratefull if you would propose me a conterexample. If this claim is true, how can I prove it?

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