What integration rule to use?

Question

I am trying to solve a differential equation, but I can't solve an integral, because I forgot which rule to apply. What rule do I use to do $$\int \frac{1}{3y-y^2}\mathrm dy\ ?$$

Answer 1

You can use Partial Fraction Decomposition since $3y-y^2=y(3-y)$. So $\frac{1}{y(3-y)}=\frac{1/3}{y}+\frac{1/3}{(3-y)}$ and certainly you know the integral of $1/y$.

I hope I could help you :)

Answer 2

Hint. By partial fraction decomposition, you have $$ \frac{3}{3y-y^2}=\frac{1}{y}-\frac{1}{y-3} $$ then integrate each part.

Answer 3

Hint: Make use of the fact that $$\frac1{3y - y^2} = \frac1{3y} - \frac1{3(y - 3)}$$ by partial fractions. Then use the linearity of the integral.