Are there any conditions of integration? When we differentiate a function $f(x)$, there are conditions under which the derivative would not exist and cannot become differentiable. However, I have tried looking online for any conditions for integration and I haven't found anything.
Are there any cases where $F(x)$ does not exist from $\int f(x)dx$? In other words, what makes a function non-integrable?
 A: 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. 
A: 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. 
A: 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.
A: You're asking when $$\int\limits_E f(x)dx$$ exists, where $E$ is some subset of the real line.  This depends on your definition of integral. For example, take $f(x) = x$.  One way to interpret $$\int_{-\infty}^{\infty} xdx$$ is as $$\lim\limits_{a \to \infty} \int_{-a}^a xdx$$ and this is clearly $0$.  But there are other ways to interpret this integral and have it not converge.  For example, if you let the positive bound to go infinity faster than the negative bound: $$\lim\limits_{a \to \infty} \int_{-a}^{2a} xdx$$  Actually, you can modify how the positive and negative bounds go to infinity and make that integral come out to whatever you want.  
On the other hand, the Lebesgue integral $\int\limits_{(-\infty,\infty)} f(x)$ is defined quite differently and is not open to interpretation.  It agrees with the Riemann integral on bounded intervals on most nice functions, but by definition $\int\limits_{(-\infty,\infty)} f$ exists if and only if when you only integrate on parts where $f$ is positive, and when you integrate on the parts where $f$ is negative, each of those converges.  So $$\int\limits_{(-\infty,\infty)} xdx$$ is not defined as a Lebesgue integral.
