# Is there a closed form for $\int x^n e^{cx}\,\mathrm dx$?

Wikipedia gives this evaluation:

$$\int x^ne^{cx}\,\mathrm dx=\frac1cx^ne^{cx}-\frac nc\int x^{n-1}e^{cx}\,\mathrm dx=\left(\frac{\partial}{\partial c}\right)^n\frac{e^{cx}}{c}$$

But I have no idea how I should exactly understand the partial part: $\left(\frac{\partial}{\partial c}\right)\frac{e^{cx}}{c}$

EDIT

Thanks for your responses so far. I should add that $n$ is not necessarily an integer. Can be for example $n = 1.2$. I'll see how far I get on learning about fractional derivatives.

• If you use the add link (but I'm not sure you have the rep yet) you can link to the source of your equation. You click on the chain icon and it opens a box to put in the URL, after which you can type in some descriptive text. Commented Feb 11, 2011 at 15:42

It means you differentiate with respect to c, n times

Use Wolfram Online Integrator, for example. The general answer is given in terms of the Incomplete Gamma Function.

• Thanks a lot! I was expecting to see a gamma function somewhere in the solution, but I never heard about this incomplete gamma function before. Commented Feb 13, 2011 at 20:00
• Note that there are two varieties of the incomplete gamma function: the upper and the lower (see also en.wikipedia.org/wiki/Incomplete_gamma_function). Commented Feb 13, 2011 at 20:06
• And just as an extra reference, a similar function is available in R, which is the language I'm using for my model, so I can calculate my integral without much trouble. rss.acs.unt.edu/Rdoc/library/stats/html/GammaDist.html Commented Feb 15, 2011 at 7:36
• @johanvdw: Indeed, the Gamma distribution and the (lower) incomplete gamma function are very closely related. Commented Feb 15, 2011 at 8:01
• I can not edit my comment, but a working link is this: stat.ethz.ch/R-manual/R-devel/library/stats/html/GammaDist.html Commented Apr 12, 2016 at 9:43

It is the (n-fold because of the exponent) derivative of $\frac{e^{cx}}{c}$ with respect to $c$, considering $x$ to be fixed. So for $n=1$ it is $\frac{c^2e^{cx}-e^{cx}}{c^2}$

• For clarity I've added to my question that n is not necessarily an integer Commented Feb 11, 2011 at 14:55