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Write a Integral recursive

  • Thread starter gop
  • Start date
gop
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1. Homework Statement

Find a recursive formula for

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

2. Homework 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.
 

Answers and Replies

rock.freak667
Homework Helper
6,230
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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
 
gop
58
0
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
 
rock.freak667
Homework Helper
6,230
31
[tex]cos^2x+sin^2x=1[/tex]
 

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