If $y^2=3x^2+2x+1$ and integration $I_n$ is defined as $I_n= \int \frac{x^n}{y}dx$, where $AI_{10}+BI_{9}+CI_{8}=x^9y$, then find the values of $A,B,C$.
I did generate the reduction formula but I am not getting any kind of given relation between the three integrations. I think my reduction formula approach is not good. How should I apply Integration by parts for reduction and how do I simplify after that. Thanks.
$(x^n y)'=n x^{n-1}y+x^n y'$ with $y=\frac{3x^2+2x+1}{y}$ and $y'=\frac{3x+1}{y}$.
$I_n'=\frac{x^n}{y}$
=> $A\frac{x^{10}}{y}+B\frac{x^9}{y}+C\frac{x^8}{y}= 9 x^8\frac{3x^2+2x+1}{y}+x^9 \frac{3x+1}{y}$
Compare the coefficients of $\frac{x^8}{y}$, $\frac{x^9}{y}$ and $\frac{x^{10}}{y}$.