Question

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

Answer 1

$$\begin{gathered} \mathrm{Note}:\mathrm{Here},\mathrm{we}\mathrm{are}\mathrm{considering}\log x\mathit{}\mathrm{as}{\log }_{\mathrm{e}}x \\ \mathrm{Let}I=\int \frac{x+1}{x\left(x+\log x\right)}dx \\ \mathrm{Putting}x+\log x=t \\ \Rightarrow 1+\frac{1}{x}=\frac{dt}{dx} \\ \Rightarrow \frac{x+1}{x}dx=dt \\ \therefore I=\int \frac{1}{t}dt \\ =\log \left|t\right|+C \\ =\log \left|x+\log x\right|+C \end{gathered}$$