Integrating Recurrence Formula for x_n = n^{-1} - 7x_{n-1}

[QCM~]

Hi guys. I'm having a bit of trouble with what I thought was a simple math question.

Homework Statement

x_{n} = \int_0^1 \frac{t^n}{t+7}dt

Show that x_0=ln(8/7) and x_n = n^{-1} - 7x_{n-1}

2. The attempt at a solution

Showing x0 = ln(8/7) is a vanilla textbook log question. I'm having trouble with the second part. I am using integration by parts on the form:
\int^1_0 t \frac{t^{n-1}}{t+7}dt
and letting u=t and dv=\frac{t^{n-1}}{t+7}dt
This give:
tx_{n-1}|^1_0 - \int_0^1 x_{n-1} dt\\<br /> = x_{n-1} - \int_0^1 x_{n-1} dt

at which point I'm stuck. I'm not sure if I've used the right IBP substitution or if I'm just almost there and it's just a case of simplifying what I have into a more general case (but can't see it).

Thanks

 

[cjc0117]

I'm still trying to figure it out using IBP, but try using long division on the integrand instead to arrive at a recursive relationship for the quotient. Then integrate.

 

[Dick]

Very astute! I don't think integration by parts leads anywhere. You might change your post to just the hint, "try long division" instead of giving the whole solution. It's more subtle.

 

[cjc0117]

Thanks Dick. And yes, I suppose you're right. I just edited it.