Question

$\int \frac{2x+1}{x^2(x+1)}dx=$

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

Answer 1

The correct option is A $\log |\frac{x}{x+1}|-\frac{1}{x}+c$

$\int \frac{2x+1}{x^2(x+1)}dx$
$\frac{2x+1}{x^2(x+1)}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{(x+1)}$
$2x+1=Ax(x+1)+B(x+1)+C{x}^2$
$A+C=0$
$A+B=2$
$B=1$
$A=1;B=1;C=-1$
$\int \frac{2x+1}{x^2(x+1)}dx=\int \frac{1}{x}dx+\int \frac{1}{x^2}dx-\int \frac{1}{x+1}dx$
$=\log x-\frac{1}{x}-\log (x+1)+c$
$=\log \mid \frac{x}{x+1}\mid -\frac{1}{x}+c$