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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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This is not a duplicate of math.stackexchange.com/questions/21516/…, since in the present question $n$ is an integer. –  Shai Covo Jun 20 '11 at 14:36
    
@Shai: Then vote to reopen. –  t.b. Jun 20 '11 at 14:38
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The "answer" to the other question involves the gamma function, which I don't think is necessary when n is a positive integer. –  Tim Jun 20 '11 at 14:48
    
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. –  Marek Jun 20 '11 at 15:06
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4 Answers 4

up vote 16 down vote accepted

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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how to vote to reopen? –  ziyuang Jun 20 '11 at 17:01
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@ziyuang: You need 3000 reputation for that. As soon as you have that there is a link right below the question that says "reopen" (if the question is closed) and "close" if the question is open. @Shai: I removed all my previous comments, so you should probably delete yours as well. –  t.b. Jun 20 '11 at 17:06
    
@Theo - done... –  Shai Covo Jun 20 '11 at 17:14
    
In case you haven't noticed, the question is reopened now. –  t.b. Jun 20 '11 at 23:21
    
@Theo, thanks, I have noticed that. –  Shai Covo Jun 20 '11 at 23:32
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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 math.stackexchange.com/questions/46733/… –  Eric Naslund Jun 21 '11 at 19:04
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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.

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+1 for using the structure of the recursion. –  Did Jun 21 '11 at 5:45
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