<s>[INST] How is it possible that $\int\frac{dy}{(1+y^2)(2+y)}$ = $\frac{1}{5}\int\frac{dy}{y+2}-\int\frac{ydy}{1+y^2}+\int\frac{2dy}{1+y^2} $? Suppose we have a fraction
$$I=\int\frac{dy}{(1+y^2)(2+y)}$$
How is it possible that 
$$5I = \int\frac{dy}{y+2}-\int\frac{ydy}{1+y^2}+\int\frac{2dy}{1+y^2} ?$$
How are they using partial fractions to do this?
 [/INST] Setting
$$\frac{1}{(1+y^2)(2+y)}=\frac{A}{y+2}+\frac{By+C}{1+y^2}$$
gives
$$1=A(1+y^2)+(y+2)(By+C),$$
i.e.
$$0y^2+0y+1=(A+B)y^2+(C+2B)y+A+2C$$
Then, solve the following system : 
$$0=A+B,\quad 0=C+2B,\quad 1=A+2C$$
</s>