Question

If $\int \frac{4x+1}{x^2+3x+2}dx=2\log |x^2+3x+2|+f\left(x\right)+C$, then $f\left(x\right)$ is
(where $C$ is integration constant)

• A. $-5\log \left|\frac{x+1}{x+2}\right|$
• B. $5\log \left|\frac{x+1}{x+2}\right|$
• C. $-\frac{5}{2}\log \left|\frac{x+1}{x+2}\right|$
• D. $\frac{5}{2}\log \left|\frac{x+1}{x+2}\right|$

Answer 1

The correct option is A $-5\log \left|\frac{x+1}{x+2}\right|$

$I=\int \frac{4x+1}{x^2+3x+2}dx=\int \frac{2(2x+3)-5}{x^2+3x+2}dx=2\int \frac{2x+3}{x^2+3x+2}dx-5\int \frac{1}{x^2+3x+2}dx=2\log |x^2+3x+2|-5\int \frac{1}{x^2+3x+\left(\frac{9}{4}\right)-\left(\frac{9}{4}\right)+2}dx=2\log |x^2+3x+2|-5\int \frac{1}{{(x+\frac{3}{2})}^2-{\left(\frac{1}{2}\right)}^2}dx=2\log |x^2+3x+2|-5\cdot \frac{1}{2\left(\frac{1}{2}\right)}\log \left|\frac{x+\frac{3}{2}-\frac{1}{2}}{x+\frac{3}{2}+\frac{1}{2}}\right|+C=2\log |x^2+3x+2|-5\log \left|\frac{x+1}{x+2}\right|+C\therefore f\left(x\right)=-5\log \left|\frac{x+1}{x+2}\right|$