Question

Evaluate $\int \frac{x^2+3x}{x^2+5x+6}dx$
(where $C$ is constant of integration)

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

Answer 1

The correct option is A $x-2\ln |x+2|+C$

Let $I=\int \frac{x^2+3x}{x^2+5x+6}dx$
$\Rightarrow I=\int \frac{x(x+3)}{(x+2)(x+3)}dx=\int \frac{x}{x+2}dx$

$\Rightarrow I=\int \frac{x+2-2}{x+2}dx=\int dx-2\int \frac{dx}{x+2}$

$\Rightarrow I=x-2\ln |x+2|+C$