Integrating $e^{f(x)}$

For the case where $f(x)$ is linear, a nice $u$-substitution works. I assume you know how to integrate $\int e^xdx$? So in order to integrate a function of the form $e^{f(x)}$, let $u=f(x)$, and thus $du=f'(x)dx$, which allows you to 'solve' for $dx$ in terms of $du$. Then your original integral goes from:
$$ \int e^{f(x)}dx $$ to $$ \int \frac{e^u}{f'(x)}du. $$ Of course, this is not always so easy to integrate, as Moron points out. When $f(x)$ is linear, you have a nice situation, because $f'(x)$ is just a constant. Other situations may not be so easily handled, as far as I'm aware.

If you mean a way to obtain an anti-derivative in terms of elementary functions, there is no such general algorithm: it is known that for $f(x) = -x^2$, $\int e^{f(x)}$ cannot be written in terms of elementary functions.

There are some general algorithms for computing anti-derivatives though, for instance: Risch's algorithm.

Your specific case is much easier than what you have generalized your problem too.

Hint: What is the derivative of $e^{-3x}$ ?