I was deriving an upper-bound of a function of $X$ which follows a density $f$.
For $p>0$,
\begin{align*} E[|X|^pI\{n< X<n+1\}]=& p\int^{n+1}_nt^{p-1}P[|X|>t]dt\\ \le&p\int^{n+1}_nt^{p-1}\frac{E[|X|^{p-1}]}{t^{p-1}}dt=pE[|X|^{p-1}] \end{align*}
I feel doubtful about this derivation; but, I cannot exactly identify and correct the part I messed up.
My guess is that the modification of $E[|X|^p]= p\int t^{p-1}P[|X|>t]dt$ for the first equality isn't a right one. It would be appreciative if I can get some comments.
The correct equality is
$$E[|X|^p I_{\{n < X < n+1\}}] = \int_0^\infty \mathbb{P}(|X|^p I_{\{x < X < n+1\}}> t) dt$$
I don't see a way to bring the indicator function outside.
Rather, I would do the following:
$$E[|X|^p I_{\{n < X < n+1\}}] = \int_{\{n < X < n+1\}} |X|^p d \mathbb{P} \leq (n+1)^p\mathbb{P}(\{n<X<n+1\})$$
If this is not a good enough bound you can try to use the density function:
$$E[|X|^p I_{\{n < X < n+1\}}] = \int_n^{n+1} x^p f(x) dx$$