Question

The value of $\int \frac{3x^2+7}{3x^3+6x^2+7x+14}dx$ is
(where $C$ is constant of integration)

• A. $\ln |x|+C$
• B. $\ln |x+1|+C$
• C. $\ln |x+2|+C$
• D. $\frac{\ln |2x+1|}{2}+C$

Answer 1

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

Let $I=\int \frac{3x^2+7}{3x^3+6x^2+7x+14}dx$
$\Rightarrow I=\int \frac{3x^2+7}{3x^2(x+2)+7(x+2)}$
$\Rightarrow I=\int \frac{3x^2+7}{(3x^2+7)(x+2)}dx=\int \frac{dx}{x+2}$

$\therefore I=\ln |x+2|+C$