Question

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

Answer 1

` Note: "Here, we are considering " log x as log_e x `
\[\text{Let I} = \int\frac{x + 1}{x\left( x + \ logx \right)}dx\]
\[\text{Putting}\ x + \log x = t\]
\[ \Rightarrow 1 + \frac{1}{x} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{x + 1}{x}dx = dt\]
\[ \therefore I = \int\frac{1}{t}dt\]
\[ = \text{log }\left| t \right| + C\]
\[ = \text{log }\left| x + \log x \right| + C\]