$dy/dx+yP(x)=Q(x) $
The standard way to solve this 1st order linear ODE is to multiply both sides by $e^{\int(P(x)}$ and then solve.
But, how to solve this type of ODEs when $P(x)$ is itself non-integrable in terms of elementary functions? For example, take the equation $dy/dx+ e^{sin(x)}y=Q(x) $. Here we have to multiply both sides with $e^{\int e^{sin(x)}dx}$. But, $e^{sin(x)}$ isn't integrable in terms of elementary functions.
One trick that comes to my mind is the Taylor series approximation.
Taylor series approximation of $e^{sin(x)}$ = $1+\frac{sin(x)}{1!} +\frac{sin^2(x)}{2!}+... $ can be used to approximately solve this particular ODE ( neglecting the higher order terms ).
Are there any other trick to solve this types of ODE?
It depends how you want your solution to look.
1) If you want a sum of integrals, expand $e^x$ as a power series: $e^x = \sum x^k/k! = \sum (\int P(x)dx)^k/k!$
If you want a product of exponentials, expand $\int P$ as a power series: $e^{\sum p_k x^k} = \prod e^{p_k x^k}$
If you want a double sum, expand both: $e^x = \sum x^k/k! = \sum(\sum p_nx^n)^k/k!$
If $P=Q$ then $\int Pe^{\int P} = e^{\int P}$.
If the integral is a specific non-elementary function (Bessel, Gamma, Hypergeometric, etc.), then it may be beneficial to write it in that form.
In some cases, doing one or more of these tricks will simplify it down to something a little nicer looking, but in the end they are all different looks to the same solution. The important thing is what you hope to get out of solving the ODE. You can use each of these forms as a potential pathway to learn something new about the solution, and if the ODE is interesting enough and the solutions aren't covered by other special functions, then generally the solution is given a name and status as a special function -- this is how Bessel, Airy, etc. came about, and coming up with different forms like this is how we know so much about them. If you have an opportunity to take a class on special functions, it's highly recommended, as you'll see all of these techniques and many more being put to use in creative and interesting ways.
You can also solve your equation numerically to get a graph of the solution curves to get an idea of what kinds of properties the solution should have. You may not suspect your function to be periodic, for example, until you see a graph which looks like it's repeating. You may also take a nonlinear dynamics approach to see where the solution approach in the long-term for different initial values.