# Write a Integral recursive

1. Homework Statement

Find a recursive formula for

$$I_{n}:=\int_{0}^{\infty}\sin^{n}(x)\cdot e^{-x}\ dx$$

2. Homework Equations

3. The Attempt at a Solution

I wrote $$I_{n+1}$$ 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.

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rock.freak667
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

I did that and got something that is partially recursive

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

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
$$cos^2x+sin^2x=1$$