First question:
I try to use partial fraction to separate the rational function ( I don't know if I am in the right direction). However, I am still stuck.
Second quesiton:
I need to find the area between the loop of the function $$y^2=x^2(x+3)$$ My strategy is to make it $y=\sqrt{x^2(x+3)}$ , find the integral of this and then multiply by two. However, I cannot find the integral.
For the first one, rewrite: $$\frac{81}{(x+1)(x^2-2x+6)} = \frac{9}{x+1} + \frac{9(x-1)}{x^2-2x+6} - \frac{18}{x^2-2x+6}$$ Then the integration should be easy.
For the second one, your integral is: $\int_{-3}^0 2 \sqrt{x(x^2 + 3)}$. Use substitution, $u = \sqrt{x+3}, du = \frac{1}{2\sqrt{x+3}}$, then your new integral is: $$4\int_0^{\sqrt{3}} (u^4 - 3u^2) du$$