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