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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.

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    If you use the $a$dd link ($b$ut I'm $n$ot 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.2011-02-11

3 Answers 3

2

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

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    I can not edit my comment, but a working link is this: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/GammaDist.html2016-04-12
3

It means you differentiate with respect to c, n times

2

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

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    For clarity I've added to my question that n is not necessarily an integer2011-02-11