Are there any conditions of integration?

A bounded function $f:[a,b]\subset\mathbb{R}\to\mathbb{R}$ is integrable if it is continuous. Actually, $f$ only needs to be almost continuous, meaning it can be discontinuous at countably-many points and still be integrable.

An interesting result due to Darboux is that any derivative has the Intermediate Value Property. As a consequence, if $f$ does not have the Intermediate Value Property, then $f$ cannot have an everywhere defined antiderivative $F$.

Remark: Your question asked about the indefinite integral. However, the notion of primary interest is the definite integral. For most mathematical purposes, the other answers are the useful ones.

The function $f:[0;1]\to \mathbb R$ defined by $$f(x) = \begin{cases}0 \mathrm{\;if\;}x\notin\mathbb Q\\1 \mathrm{\;if\;}x\in\mathbb Q\end{cases}$$ is not Riemann-integrable : see here.