Question

\[\int\limits_1^2 \frac{x}{\left( x + 1 \right) \left( x + 2 \right)} dx\]

Answer 1

\[Let\ I = \int_1^2 \frac{x}{\left( x + 1 \right)\left( x + 2 \right)} d\ x\ . Then, \]
\[I = \int_1^2 \left( \frac{- 1}{\left( x + 1 \right)} + \frac{2}{\left( x + 2 \right)} \right) d x\]
\[ \Rightarrow I = - \int_1^2 \frac{1}{\left( x + 1 \right)} dx + 2 \int_1^2 \frac{1}{\left( x + 2 \right)} dx\]
\[ \Rightarrow I = \left[ - \log \left( x + 1 \right) + 2 \log \left( x + 2 \right) \right]_1^2 \]
\[ \Rightarrow I = - \log 3 + 2 \log 4 + \log 2 - 2 \log 3\]
\[ \Rightarrow I = 5 \log 2 - 3 \log 3\]
\[ \Rightarrow I = \log 2^5 - \log 3^3 \]
\[ \Rightarrow I = \log \frac{32}{27}\]