Question

$\underset{1}{\overset{2}{\int }}\frac{x}{(x+2)(x+3)}dx$ is equal to

• A. $\log \frac{125}{4096}$
• B. $\log \frac{1125}{1024}$
• C. $\log \frac{125}{512}$
• D. $\log \frac{1125}{1024}$

Answer 1

The correct option is D $\log \frac{1125}{1024}$

$\underset{1}{\overset{2}{\int }}\frac{x}{(x+2)(x+3)}dx$
Using partial fraction we get,
$\frac{x}{(x+2)(x+3)}=\frac{-2}{x+2}+\frac{3}{x+3}$
$=\underset{1}{\overset{2}{\int }}\frac{-2}{x+2}dx+\underset{1}{\overset{2}{\int }}\frac{3}{x+3}dx$
$=[-2\log |x+2|+3\log |x+3|{]}_1^2=(-2\log 4+3\log 5)-(-2\log 3+3\log 4)=3\log 5+2\log 3-5\log 4=\log \frac{125\times 9}{1024}=\log \frac{1125}{1024}$