Question

Evaluate : $\int \frac{1}{x(1+\log x)}dx$

Answer 1

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

Let (1 + log x) = t

$$\begin{gathered} {\displaystyle \mathrm{or}}{\displaystyle ,}{\displaystyle }{\displaystyle \frac{1}{x}}dx=dt \\ \Rightarrow I=\int {\displaystyle \frac{1}{t}}dt \end{gathered}$$

$$\begin{gathered} \Rightarrow I=\log \left|t\right|+C \\ \therefore I=\log \left|1+\log x\right|+C \end{gathered}$$