Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_1^2\frac{x}{\left(x+1\right)\left(x+2\right)}\mathit{d}x\mathit{.}\mathrm{Then}, \\ I={\int }_1^2\left(\frac{-1}{\left(x+1\right)}+\frac{2}{\left(x+2\right)}\right)\mathit{d}x \\ \Rightarrow I\mathit{=}\mathit{-}{\int }_1^2\frac{\mathit{1}}{\left(x+1\right)}\mathit{}dx+2{\int }_1^2\frac{1}{\left(x+2\right)}dx \\ \Rightarrow I={\left[-\log \left(x+1\right)+2\log \left(x+2\right)\right]}_1^2 \\ \Rightarrow I=-\log 3+2\log 4+\log 2-2\log 3 \\ \Rightarrow I=5\log 2-3\log 3 \\ \Rightarrow I=\log 2^5-\log 3^3 \\ \Rightarrow I=\log \frac{32}{27} \end{gathered}$$