Question

The value of $\int \frac{2x}{x^2+3x+2}dx$ is
(where $C$ is integration constant)

• A. $\ln |x-2|+2\ln |x+1|+C$
• B. $2\ln |x+2|+4\ln |x+1|+C$
• C. $\ln |x+2|+2\ln |x-1|+C$
• D. $4\ln |x+2|-2\ln |x+1|+C$

Answer 1

The correct option is D $4\ln |x+2|-2\ln |x+1|+C$

$\frac{2x}{x^2+3x+2}=\frac{2x}{(x+1)(x+2)}\Rightarrow \frac{2x}{(x+1)(x+2)}=\frac{A}{(x+1)}+\frac{B}{(x+2)}\Rightarrow 2x=A(x+2)+B(x+1)\cdots \left(1\right)$

Equating the coefficients of $x$ and constant term, we obtain
$A+B=22A+B=0$
Solving these equations, we obtain
$A=-2,B=4\therefore \int \frac{2x}{(x+1)(x+2)}dx=\int [\frac{4}{(x+2)}-\frac{2}{(x+1)}]dx=4\ln |x+2|-2\ln |x+1|+C$