Note: You can get full credit for this problem by just entering the final answer $($to the last question$)$ correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.

Consider the indefinite integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{8x^3+5x^2+45x+27}{x^4+9x^2}dx$

Then the integrand has partial fractions decomposition

$\frac{a}{x^2}+\frac{b}{x}+\frac{cx+d}{x^2+9}$

where

$a=3$

$b=5$

$c=3$

$d=$

Integrating term by term, we obtain that

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{8x^3+5x^2+45x+27}{x^4+9x^2}dx=$

$5ln(|x|)-\frac{3}{x}+\frac{3}{2}(ln(x^2+9))+\frac{2arctan(\frac{x}{3})}{3}+C$