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

Question

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. — Ross Millikan, Feb 11 '11 at 15:42

score 4 · Answer 1 · answered Feb 11 '11 at 14:06

4

It means you differentiate with respect to c, n times

answered Feb 11 '11 at 14:06

TROLLHUNTER

8,875

score 3 · Answer 2 · answered Feb 11 '11 at 14:10

3

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 $\frac{c^2e^{cx}-e^{cx}}{c^2}$

answered Feb 11 '11 at 14:10

Ross Millikan

383,099

For clarity I've added to my question that n is not necessarily an integer – johanvdw Feb 11 '11 at 14:55

score 3 · Accepted Answer · answered Feb 13 '11 at 10:02

3

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

answered Feb 13 '11 at 10:02

Shai Covo

24,383

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. – johanvdw Feb 13 '11 at 20:00
1

Note that there are two varieties of the incomplete gamma function: the upper and the lower (see also http://en.wikipedia.org/wiki/Incomplete_gamma_function). – Shai Covo Feb 13 '11 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. http://rss.acs.unt.edu/Rdoc/library/stats/html/GammaDist.html – johanvdw Feb 15 '11 at 07:36
1

@johanvdw: Indeed, the Gamma distribution and the (lower) incomplete gamma function are very closely related. – Shai Covo Feb 15 '11 at 08:01
I can not edit my comment, but a working link is this: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/GammaDist.html – johanvdw Apr 12 '16 at 09:43
@ShaiCovo but here the power of euler's number is not negative – Shimura Variety Jun 03 '22 at 03:12

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

3 Answers3

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