Question

Find the integral of $\int \frac{dx}{(x+1)(x+2)}$

• A. $\log \frac{x+2}{x+1}+C$
• B. $\log (x+1)+\log (x+2)+C$
• C. $\log \frac{x+1}{x+2}+C$
• D. $\log \frac{x}{x+2}+C$

Answer 1

The correct option is C $\log \frac{x+1}{x+2}+C$

Let
$\frac{1}{(x+1)(x+2)}=\frac{A}{x+1}+\frac{B}{x+2}$
$1=A(x+2)+B(x+1)\Rightarrow Ax+Bx+2A+B=1$
Comparing the coefficient of $x$ and constant on both sides,
$A+B=0$ and $2A+B=1$
Solving the above two equations, we get
$A=1$ and $B=-1$
$\Rightarrow \int \frac{dx}{(x+1)(x+2)}=\int (\frac{1}{(x+1)}-\frac{1}{(x+2)})dx$
$=\log (x+1)–\log (x+2)+C=\log \frac{x+1}{x+2}+C$