Norm $L^p$ for the function $f(x) = \sin(e^{x^3}) 1_{A \cap \mathbb{Q}}(x) + e^{-x} 1_{A/\mathbb{Q}}(x)$

Question

Let $A = [0,+\infty[$. For all $p \in ]1,\infty[$, I have to calculate the norm $||.||_p$ of the function $f : A \rightarrow \mathbb{R}$ defined by :

$f(x) = \sin(e^{x^3}) 1_{A \cap \mathbb{Q}}(x) + e^{-x} 1_{A/\mathbb{Q}}(x)$.

I wrote to begin, $f(x) = \begin{cases} \sin(e^{x^3}) & \text{ if } x \in \mathbb{R}^+ \cap \mathbb{Q} \\ e^{-x} & \text{ if } x \in \mathbb{R}^+ / \mathbb{Q} \end{cases}$.

Then, I wrote $||f||_p^p = \int |f(x)|^p dx = \int \sin^p(e^{x^3}) 1_{A \cap \mathbb{Q}}(x) dx + \int e^{-px} 1_{A/\mathbb{Q}}(x) dx$. But I don't know how to calculate this integral.

The first part is very hard to calculate and I'm not sure it's just. The second part is easier to calculate but what are the upper bound and the lower bound ? Thank you in advance for help, I'm beginner in spaces $L^p$.

Answer 1

The first part is zero. Think of where the function is nonzero.