Question

Evaluate : $\int \frac{(4x+1)dx}{x^2+3x+2}$

• A. $=2log|x^2+5x+2|-5log\left|\frac{x+1}{x+2}\right|+C$
• B. $=2log|x^2+3x+2|-5log\left|\frac{x+3}{x+2}\right|+C$
• C. $=2log|x^2+3x+2|-5log\left|\frac{x+1}{x+2}\right|+C$
• D. $=2log|x^2+3x+2|-5log\left|\frac{x+1}{x+3}\right|+C$

Answer 1

The correct option is C $=2log|x^2+3x+2|-5log\left|\frac{x+1}{x+2}\right|+C$

$I=\int \frac{4x+1}{x^2+3x+2}dx$
$=\int \frac{2(2x+3)-5}{x^2+3x+2}dx$
$=2\int \frac{2x+3}{x^2+3x+2}dx-5\int \frac{1}{x^2+3x+2}dx$

Put $x^2+3x+2=t$

$\Rightarrow (2x+3)dx=dt$
$=2\int \frac{1}{t}dt-5\int \frac{1}{x^2+3x+\left(9/4\right)-\left(9/4\right)+2}dx$
$=2log|x^2+3+2|-5\int \frac{1}{{(x+3/2)}^2-{\left(1/2\right)}^2}dx$

Put $(x+3/2)=u$

$\Rightarrow dx=du$
$=2log|x^2+3x+2|-5\int \frac{1}{u^2-{\left(1/2\right)}^2}dx$
$=2log|x^2+3x+2|-5\times \frac{1}{2\left(1/2\right)}log\left|\frac{x+\frac{3}{2}-\frac{1}{2}}{x+\frac{3}{2}+\frac{1}{2}}\right|+C$
$=2log|x^2+3x+2|-5log\left|\frac{x+1}{x+2}\right|+C$