Let $X$ be a real-valued continuous random variable with CDF $F(x)$, and let $k \in \mathbb{R}$ be a constant. I'm interested in the expected value of the truncated distribution $X \, | \, X < k$.
How can we further simply the conditional expectation $\operatorname{E}[X \,|\, X < k]$ in terms of $F(x)$? We can write the following:
$$ \operatorname{E}[X \,|\, X < k] = \int_{-\infty}^k x\frac{F'(x)}{F(k)} dx = \frac{1}{F(k)}\int_{-\infty}^k x F'(x) dx$$
Can you simplify this further to get rid of the integral for a general $F$ (rather than just, e.g. $X \sim \mathcal{N}(\mu, \sigma^2)$, discussed here)?
Computing your integral is at least as hard as computing the mean of $X$, given by $$E[X]=\int_{-\infty}^\infty x F'(x)dx. \quad (1)$$
If you wouldn't write the mean of a random variable for general $F$ any simpler than in $(1)$, you should consider your expression to be simplified.