TITLE: Integrating $e^{f(x)}$
QUESTION [6 upvotes]: can someone tell me a way of integrating functions like $e^{f(x)}$
I have a specific case: $\int e^{-3x}\,\mathrm{d}x$
PS: I'm not looking for the answer of this, but the way of doing it.
Thanks for your help

REPLY [5 votes]: 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.