Asymptotic bound on the expected value with a bound on the tail probability

Question

Let $X_n$ be a sequence of positive random variables, I have a tail bound of the form $$ \mathbb{P}(X_n \geq \delta_n) \leq \gamma_n $$ with $\delta_n \rightarrow 0$ and $\gamma_n\rightarrow 0$ as $n\rightarrow \infty$. I need to find a bound on $\mathbb{E}(X_n)$ for $n\rightarrow \infty$. Intuitively, I would use the following identity $$ \mathbb{E}(X_n) = \int^{\infty}_0 \mathbb{P}(X_n > x) dx, $$ but I can't actually move forward. Any suggestions?