Question

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

Answer 1

\[Let\ I = \int_1^2 \frac{1}{x \left( 1 + \log x \right)^2} d x . Then, \]
\[Let\ \left( 1 + \log x \right) = t . Then, \frac{1}{x} dx = dt\]
\[When\ x = 1, t = 1\ and\ x\ = 2, t = 1 + \log 2\]
\[ \therefore I = \int_1^\left( 1 + \log 2 \right) \frac{1}{t^2} dt\]
\[ \Rightarrow I = \left[ \frac{- 1}{t} \right]_1^\left( 1 + \log 2 \right) \]
\[ \Rightarrow I = - \frac{1}{1 + \log 2} + 1\]
\[ \Rightarrow I = \frac{\log 2}{1 + \log 2}\]