Question

Integrate:
$\frac{3x+5}{(x+1)(x-2{)}^2}dx$

Answer 1

Let $\frac{3x+5}{(x+1){(x-2)}^2}=\frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{{(x-2)}^2}$

On comparing, we get

$A=\frac{2}{9}$, $B=\frac{-2}{9}$, $C=\frac{11}{3}$

So, $\int \frac{(3x+5)}{(x+1){(x-2)}^2}dx=\int \frac{2}{9(x+1)}dx+\int \frac{-2}{9(x-2)}dx+\int \frac{11}{3{(x-2)}^2}dx$

So, $\int \frac{3x+5}{(x+1){(x-2)}^2}=\frac{2}{9}log|x+1|-\frac{2}{9}ln(x-2)-\frac{11}{3}\frac{1}{(x-2)}+C$