Question

Integrate the rational function: $\frac{2x}{x^2+3x+2}$

Answer 1

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

2x = A(x+2)+B(x+1) ...(1)

Substituting x = −1 and −2 in equation (1), we obtain

A = −2 and B = 4

∴ $\frac{2x}{(x+1)(x+2)}=\frac{-2}{(x+1)}+\frac{4}{(x+2)}$)

\Rightarrow \int\frac{2x}{(x+1)(x+2)}dx=\int\bigg\\{\frac{4}{(x+2)}-\frac{2}{(x+1)}\bigg\\}dx

= 4log|x+2|-2log|x+1|+C