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Write a Integral recursive

  1. Apr 2, 2008 #1

    gop

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    1. The problem statement, all variables and given/known data

    Find a recursive formula for

    [tex]I_{n}:=\int_{0}^{\infty}\sin^{n}(x)\cdot e^{-x}\ dx[/tex]

    2. Relevant equations



    3. The attempt at a solution

    I wrote [tex]I_{n+1}[/tex] and tried to integrate by parts and with substitution, however, I wasn't able to get a the original term so that I could write it recursively.
     
  2. jcsd
  3. Apr 2, 2008 #2

    rock.freak667

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    Homework Helper

    You need to do integration by parts twice and then remember that to pick the trig terms as 'u' when integrating by parts the second time
     
  4. Apr 2, 2008 #3

    gop

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    I did that and got something that is partially recursive

    [tex]I_{n}=n^{2}\cdot\int_{0}^{\infty}\sin^{n-2}(x)\cdot\cos^{2}(x)\cdot e^{-x}\ dx-n\cdot I_{n-2}[/tex]

    However, I don't know how to eliminate the cos^2 at all. Moreover, one should prove with the recursive function that when n->infinity I_n->0. But with this result I don't really know how to tackle this problem either. Somehow I'm totally stuck with this one.

    thx
     
  5. Apr 3, 2008 #4

    rock.freak667

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    Homework Helper

    [tex]cos^2x+sin^2x=1[/tex]
     
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