Question

Evaluate $\int \frac{2x+3}{x^2-4x-21}dx$

(where $C$ is constant of integration)

• A. $\frac{17}{10}\ln |x-7|+\frac{13}{10}\ln |x+3|+C$
• B. $\frac{17}{10}\ln |x-7|+\frac{3}{10}\ln |x+3|+C$
• C. $\frac{13}{10}\ln |x+7|+\frac{15}{4}\ln |x-3|+C$
• D. $\frac{13}{10}\ln |x-7|+\frac{3}{10}\ln |x+3|+C$

Answer 1

The correct option is B $\frac{17}{10}\ln |x-7|+\frac{3}{10}\ln |x+3|+C$

Let $I=\int \frac{2x+3}{x^2-4x-21}dx$
$\Rightarrow I=\int \frac{2x+3}{(x-7)(x+3)}dx$
Using partial fraction, we get

$\frac{2x+3}{(x-7)(x+3)}=\frac{A}{x-7}+\frac{B}{x+3}$

$\Rightarrow A=\frac{17}{10},B=\frac{3}{10}$

$\Rightarrow I=\frac{17}{10}\int \frac{dx}{x-7}+\frac{3}{10}\int \frac{dx}{x+3}$

$\therefore I=\frac{17}{10}\ln |x-7|+\frac{3}{10}\ln |x+3|+C$