Question

Integrate the rational function: \(\frac{x}{(x+1)(x+2)}\)

Answer 1

Let  \(\frac{x}{(x+1)(x-+2)} = \frac{A}{(x+1)}+\frac{B}{(x+2)}\)
\(\Rightarrow x =  A(x+2)+B(x+1)\)
Equating the coefficients of x and constant term, we obtain
A + B = 1
2A + B = 0
On solving, we obtain
A = −1 and B = 2

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

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

= \(- \log \mid x+1 \mid +2\log \mid x+2 \mid+C\)

= \(\log(x+2)^2 -\log \mid x+1 \mid+C\)

= \(\log(\frac{x+2)^2}{ (x+1)}+C\)