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)
Hello,
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