Let $I=\int \frac{2x}{x^2+3x+2}dx$ Let $\frac{2x}{x^2+3x+2}$ $=\frac{2x}{(x+1)(x+2)}$ $=\frac{A}{x+1}+\frac{B}{x+2}$ ⇒ 2x = A(x + 2) + B(x + 1) ...(i) Putting x = – 1 in (i), we get 2 (–1) = A(–1 + 2) ⇒ A = – 2 Putting x = – 2 in (i), we get 2 (–2) = B(–2 + 1) ⇒ B = 4 $\therefore \frac{2x}{x^2+3x+2}$ $=\frac{-2}{x+1}+\frac{4}{x+2}$ $\therefore \int \frac{2x}{x^2+3x+2}dx$ $=-2\int \frac{1}{x+1}dx+4\int \frac{1}{x+2}dx$$=-2\log|x+1|+4\log|x+2|+C$