Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $X$ be a random variable defined on $\Omega$. Given any event $A \in \mathcal{F}$ with positive probability, the expectation of $X$ conditioned on $A$ is defined as $$E(X|A) := \frac{E(X1_{A})}{\mathbb{P}(A)} = \frac{1}{\mathbb{P}(A)}\int_{\Omega}X1_{A}d\mathbb{P}.$$
My question is the following: is there an easy way to compute this integral, regardless of the event $A$? If $X$ is discrete, the computation is clear. If $X$ is absolutely continuous and $A = X^{-1}(B)$ for a given borel set $B \in \mathcal{B}$, it makes sense to define the conditional density function as $$f_{X|A} := \frac{f_{X}1_{B}}{\mathbb{P}(A)},$$ hence $$E(X|A) := \int_{\mathbb{R}}xf_{X|A}dx.$$
What can we say for a general event $A \in \mathcal{F}$?