I tried using integration by parts but it just seems to get more complicated regardless of which function I set for f(x) or g(x). Here is the integral:
$\displaystyle \int_0^{\infty}x^2e^{-x(y+1)}dx$
Here is one way to tackle it.
$\displaystyle \int x^{2}e^{-x(y+1)}dx$
Since the e term is a product of a polynomial, x^2, we can try this:
$\displaystyle \frac{d}{dx}[ax^{2}+bx+c]e^{-x(y+1)}=x^{2}e^{-x(y+1)}$..............[1]
Use the product rule on the left side and get:
$\displaystyle (-ay-a)x^{2}+x(-by+2a-b)-cy+b-c=x^{2}e^{-x(y+1)}$
Equate coefficients:
$\displaystyle -ay-a=1$
$\displaystyle -by+2a-b=0$
$\displaystyle -cy+b-c=0$
Solving the system results in:
$\displaystyle a=\frac{-1}{y+1}, \;\ b=\frac{-2}{(y+1)^{2}}, \;\ c=\frac{-2}{(y+1)^{3}}$
Sub them back into [1]:
$\displaystyle \frac{d}{dx}\left[\frac{-x^{2}}{y+1}-\frac{2x}{(y+1)^{2}}-\frac{2}{(y+1)^{3}}\right]e^{-x(y+1)}=x^{2}e^{-x(y+1)}$
Integrate both sides and get:
$\displaystyle \left(\frac{-x^{2}}{y+1}-\frac{2x}{(y+1)^{2}}-\frac{2}{(y+1)^{3}}\right)e^{-x(y+1)}=\int x^{2}e^{-x(y+1)}dx$
There she is.