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 June 6th, 2010, 12:36 PM #1 Newbie   Joined: Jun 2010 Posts: 2 Thanks: 0 Help with limits for calculus homework Evaluate the integral for n = 0, 1, 2, and 3. http://www.webassign.net/cgi-bin/symima ... 29%29%20dx I've already solved the one for n=0, it's 10. not sure how to exactly go about on the other ones.
 June 6th, 2010, 01:25 PM #2 Global Moderator   Joined: May 2007 Posts: 6,821 Thanks: 723 Re: Help with limits for calculus homework Integration by parts can give you the integral involving x^n in terms of the integral involving x^(n-1). Start with n=1 and work your way up.
 June 6th, 2010, 01:45 PM #3 Newbie   Joined: Jun 2010 Posts: 2 Thanks: 0 Re: Help with limits for calculus homework I've tried. but I keep getting answers that don't make sense. I've taken the the integral from 0 to inf, and at n=1. I've let u=x, dv=e^-x and so on. However, my answer is still wrong.
 June 6th, 2010, 04:13 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond Re: Help with limits for calculus homework I believe I have an answer for n = 1, if it's any help: $\int10xe^{-x}\,dx$ Integration by parts: Let $u\,=\,10x,\,du\,=\,10\,dx,\,dv\,=\,e^x\,dx,\,v\,=\ ,-e^{-x}$ $-10xe^{-x}\,-\,\int-10e^{-x}\,dx$ $=\,-10xe^{-x}\,+\,\int10e^{-x}\,dx$ $\,=-10xe^{-x}\,-\,10e^{-x}$ (We're doing a definite integral so I'll leave out the constant of integration). If my intuition is correct, to evaluate the definite integral we take the limit: $\lim_{x \to \infty}\left(-10xe^{-x}\,-\,10e^{-x}\right)\,=\,\lim_{x \to \infty}-10xe^{-x}\,-\,\lim_{x \to \infty}10e^{-x}$ $=\lim_{x \to \infty}\,-10xe^{-x}\,-\,0$ For the limit $-10\lim_{x \to \infty}xe^{-x}$ rewrite as $\frac{x}{1/e^{-x}}$ and apply L'Hopital's Rule, thus the limit is . So, $\int_{0}^{\infty}10xe^{-x}\,=\,0\,-\,(-10)\,=\,10$ I think my math is good -- hope it helped.
 June 6th, 2010, 05:02 PM #5 Global Moderator   Joined: Dec 2006 Posts: 20,966 Thanks: 2216 By the definition of the gamma function, $\int_0\,^\infty10x^ne^{-x}dx\,=\,10\text{\Gamma}(n+1).$ If n is a non-negative integer, that's 10(n!). I'll leave it to you to look up anything more you need (or use the method mathman suggested). Integration by parts: let $u\,=\,10x,\,du\,=\,10\,dx,\;dv\,=\,e^{-x}\,dx,\,v\,=\,-e^{-x}.$ $\int10xe^{-x}\,dx\,=\,-10xe^{-x}\,-\,\int-10e^{-x}\,dx\,=\,-10xe^{-x}\,-\,10e^{-x}\,+\,\text{C.}$ etc.
June 8th, 2010, 03:48 PM   #6
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 Originally Posted by skipjack By the definition of the gamma function, $\int_0\,^\infty10x^ne^{-x}dx\,=\,10\text{\Gamma}(n+1).$ If n is a non-negative integer, that's 10(n!). I'll leave it to you to look up anything more you need (or use the method mathman suggested). Integration by parts: let $u\,=\,10x,\,du\,=\,10\,dx,\;dv\,=\,e^{-x}\,dx,\,v\,=\,-e^{-x}.$ $\int10xe^{-x}\,dx\,=\,-10xe^{-x}\,-\,\int-10e^{-x}\,dx\,=\,-10xe^{-x}\,-\,10e^{-x}\,+\,\text{C.}$ etc.
Integration by parts:

$\int_0\,^\infty10x^ne^{-x}dx\,=n\int_0\,^\infty10x^{n-1}e^{-x}dx\$, etc.

 June 9th, 2010, 04:53 PM #7 Newbie   Joined: Jun 2010 Posts: 21 Thanks: 0 Re: Help with limits for calculus homework You can make this a series so that you do not have to repeat the integral. So basically if you call Un=Integral(10*x^n*e^(-x)) (n>0). You do the integral by parts of that and you will get Un=-10X^(n)*e^(-x) + nUn-1 and Uo=10 like u already figured. And all you have to do now is plug in n for each number. Sorry I don't know how to put an integral here so hope u can read it.

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