Question

$$\underset{1}{\overset{2}{\int }}\frac{x}{(x+1)(x+2)}dx$$

Answer 1

$$LetI={\int }_1^2\frac{x}{(x+1)(x+2)}dx.Then,$$
$$I={\int }_1^2(\frac{-1}{(x+1)}+\frac{2}{(x+2)})dx$$
$$\Rightarrow I=-{\int }_1^2\frac{1}{(x+1)}dx+2{\int }_1^2\frac{1}{(x+2)}dx$$
$$\Rightarrow I={[-\log (x+1)+2\log (x+2)]}_1^2$$
$$\Rightarrow I=-\log 3+2\log 4+\log 2-2\log 3$$
$$\Rightarrow I=5\log 2-3\log 3$$
$$\Rightarrow I=\log 2^5-\log 3^3$$
$$\Rightarrow I=\log \frac{32}{27}$$