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I have a question about integrating by parts for $$\int_{L}^{U}\left[x^{a} \cdot e^{-bx}\right]\,dx$$ for positive reals $L,U$ with $L<U$ ($L, U \in [0, +\infty) $). I'm interested in cases with $a$ and $b$ positive constants, either integer or non-integer.

case 1 : a and b are positive integers

case 2 : a and b are positive non-integers

Are there formulas that can give solutions for any given $a$, $b$, $U$, and $L$?

Incomplete gamma function is similar as this, but it is a special case of $b = 1$.

Any help would be appreciated.

user3601704
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  • Are $a$ and $b$ constants? – Jam Aug 13 '14 at 22:03
  • yes, a and b are given constants. – user3601704 Aug 13 '14 at 22:10
  • If possible, who can tell me how to type mat formulas here ? thanks ! – user3601704 Aug 13 '14 at 22:10
  • I've converted your question into symbolic form via Mathjax. I'm not sure it's quite right, though: did you mean $e^{a-1}$ or $e^{(a-1)x^{-b}}$? It wasn't clear from your original text. For a Mathjax tutorial, see this link. – Semiclassical Aug 13 '14 at 22:10
  • @user3601704 Then you can take $e^{a-1}$ out as a constant and integrate $x^{1-b}$ quite easily. But given that you reference the incomplete gamma function, I don't feel as though that's what you want. – Jam Aug 13 '14 at 22:11
  • It is likely intended to type something else. If the intended function has shape $x^p e^{-qx}$, one can make the change of variable $t=qx$. If you have $\exp(-qx^r)$ a change of variable will also work. – André Nicolas Aug 13 '14 at 22:12
  • I have a typo here. – user3601704 Aug 13 '14 at 22:12
  • @user3601704 Here's a formatting tutorial http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Jam Aug 13 '14 at 22:12
  • @user3601704 If $a$ & $b$ are constants then $e^{-b}$ is too & you can integrate $x^a$ as you would any polynomial: $\int ax^n,\mathrm{d}x=\frac{ax^{n+1}}{n+1}, n\neq-1$. – Jam Aug 13 '14 at 22:18
  • @EulCan, OP has been updated again, thanks ! – user3601704 Aug 13 '14 at 22:20
  • @user3601704 This page gives a solution for $L=0$, $U=+\infty$ http://math.stackexchange.com/questions/601900/how-to-solve-this-integral-int-0-infty-xa-e-bxdx . – Jam Aug 13 '14 at 22:26
  • thanks ! what if $L, U \in [0, +\infty) $ ? – user3601704 Aug 13 '14 at 22:29
  • If $L\gt 0$, then one uses the incomplete gamma function, after the change of variable $t=bx$. – André Nicolas Aug 13 '14 at 22:49

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