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How can I solve this indefinite integral for an arbitrary integer $n>0$?

$ \int{x^n e^x dx}$

I could partially integrate it for small $n$, but that's not really a solution.


Edit: (TB) This question is closely related to: Is there a closed form solution for $\int x^n e^{cx}$?, but it is more elementary, because $n$ is an integer here.

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    Why do you say it's not a solution? The answer obviously must contain all the powers of $x^k$ for $0 \leq k \leq n$ so the solution can't be much simpler than this. E.g. for x > 0 you might try $\left((-\partial_{\alpha})^n \int e^{-\alpha x}\right) \Big\vert_{\alpha = 1}$ but now you have to differentiate $n$ times so this is essentially the same thing.2011-06-20

3 Answers 3

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Hint: Use integration by parts.

EDIT: Try several values of $n$. $ \int {x e^x dx} = (x - 1)e^x + C $ $ \int {x^2 e^x dx} = (x^2 - 2x + 2)e^x + C. $ $ \int {x^3 e^x dx} = (x^3 - 3x^2 + 6x - 6)e^x + C. $ $ \int {x^4 e^x dx} = (x^4 - 4x^3 + 12x^2 - 24x + 24)e^x + C. $ $ \int {x^5 e^x dx} = (x^5 - 5x^4 + 20x^3 - 60x^2 + 120x - 120)e^x + C. $ Conclude that $ \int {x^n e^x dx} = \bigg[\sum\limits_{k = 0}^n {( - 1)^{n - k} \frac{{n!}}{{k!}}x^k } \bigg]e^x + C. $

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    @Theo, thanks, I have noticed that.2011-06-20
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You could use the generating function approach.
$ \eqalign{\int_0^X e^{tx} e^x \ dx &= \frac{e^{(1+t)X} - 1}{1+t}\cr &= \sum_{k=0}^\infty (-1)^k t^k \left(e^X -1 + \sum_{j=1}^\infty e^X \,\frac{X^j}{j!} t^j\right)\cr &= \sum_{n=1}^\infty \left((-1)^n (e^X - 1) + \sum_{j=1}^n (-1)^{n-j} \frac{X^j}{j!} e^X \right) t^n\cr}$ But also $ \int_0^X e^{tx} e^x \, dx = \sum_{n=0}^\infty \frac{t^n}{n!} \int_0^X x^n e^x \, dx$ Equating coefficients of $t^n$ from both sides, $ \int_0^X x^n e^x\, dx = (-1)^n n! (e^X - 1) + \sum_{j=1}^n (-1)^{n-j} \frac{n!}{j!} X^j e^X $

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    Interesting coincidence! We must have been thinking about this in a similar way as I also posted this answer on Mike Spivey's follow up question. (Although it is admittedly more suited as a response here) See http://math.stackexchange.com/questions/46733/is-there-a-way-to-prove-int-xn-ex-dx-ex-sum-k-0n-1k-frac/46745#467452011-06-21
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I find it a little difficult for me to guess the solution by trying several $n$. I would like to do it as following:

$\begin{align}\int x^ne^xdx&=x^ne^x+(-1)n\int x^{n-1}e^xdx,\qquad n\geq 1\\ \int x^0e^xdx&=e^x\end{align}$

Then you get the recurrence relation:

$\begin{align}a_n(x)&=x^ne^x+(-1)na_{n-1}(x),\qquad n\geq 1\\ a_0(x)&=e^x\end{align}$

With the recursive formula, it may be easier to find the pattern of the result.