# Thread: Finding a recursive formula by using integration by parts

1. ## Finding a recursive formula by using integration by parts

Does anyone know how you would go about finding the recursive formula when it is a fraction.

E.g. Find recursive formula for 1/(x^3+1)^w (dx)

2. Hello,
Originally Posted by hats_06
Does anyone know how you would go about finding the recursive formula when it is a fraction.

E.g. Find recursive formula for 1/(x^3+1)^w (dx)
Integration by parts most of the time.

Which is the case here.

$\displaystyle \left(\frac{1}{(x^3+1)^w}\right)'=\left((x^3+1)^{-w}\right)'=\frac{-3wx^2}{(x^3+1)^{w+1}}=du$

$\displaystyle dv=1 \implies v=x$

$\displaystyle I_w=\int \frac{1}{(x^3+1)^w} ~dx=\frac{x}{(x^3+1)^w}+3w \int \frac{x \cdot x^2}{(x^3+1)^{w+1}} ~dx$

$\displaystyle =\frac{x}{(x^3+1)^w}+3w \int \frac{x^3}{(x^3+1)^{w+1}} ~dx$

Write $\displaystyle x^3=(x^3+1)-1$ and remember that $\displaystyle I_{w+1}=\int \frac{1}{(x^3+1)^{w+1}} ~dx$ after that, it's just algebra and you're done

3. can you do it writing it like this?

(x^3+1)^-w (dx)

sorry man i havent done recursions for a while...does that help?

EDIT: beaten to it...i was nowhere near as much help.

4. Hey moo thanks for your help! I had gotten it right up until this point:

Write and remember that after that, it's just algebra and you're done

I dont still quite understand why

5. Oh wait i understand!! Thanks so much once again!

6. Heey its me again...I got slightly confused again when i evaluated it...but i ended up with this:

Iw = 3wIw - 3wIw+1 + (x)/(x^3+1)^w

Does that sound right?

7. Originally Posted by hats_06
Heey its me again...I got slightly confused again when i evaluated it...but i ended up with this:

Iw = 3wIw - 3wIw+1 + (x)/(x^3+1)^w

Does that sound right?
Perfect

Now, write $\displaystyle I_{w+1}$ in terms of $\displaystyle I_w$ (that's not much, it's just for aesthetics ^^)

8. So just:

Iw+1 = Iw - (1/3w)Iw + (x)/3w(x^3+1)^w

Is that it?

9. Yes it is

or more precisely (1-1/3w)I_w

10. Thanks so much my genius friend!