# Thread: Integration - Recursive Formulae

1. ## Integration - Recursive Formulae

I did Integration by Parts and ended up with:

$\displaystyle I_n = e - nI_{n-1}$

Not sure where to head from here?

Thanks.

2. ## Re: Integration - Recursive Formulae

Hey eskimogenius.

Integrating by parts gives

I_{n-1}
= [x^(n-1)*e^x]{0,1} - 1/n*Int{0,1}(x^n*e^(x)dx)
= e - 1/n*I{n}

Assuming d/dx (x^n/n) = x^(n-1).

3. ## Re: Integration - Recursive Formulae

Sorry, I think the question might have been wrong.

If you sub:
$\displaystyle n = n+1$ into the result I gave, you arrive at the answer except a + instead of - sign.

4. ## Re: Integration - Recursive Formulae

Originally Posted by eskimogenius
Sorry, I think the question might have been wrong.
If you sub: $\displaystyle n = n+1$ into the result I gave, you arrive at the answer except a + instead of - sign.
You are correct as posted there is a small typo in it.
It should be $\displaystyle {I_{n + 1}} = e - (n + 1){I_n}$.

If you start with $\displaystyle {I_n} = \int_0^1 {{x^n}{e^x}dx}$ and then use the parts decomposition $\displaystyle u=e^x~\&~dv=x^n$.

That will give the corrected formula.