Question

Write a value of $\int \frac{1}{x{\left(\log x\right)}^n}dx$.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{dx}{x{\left(\log x\right)}^n} \\ =\int \frac{{\left(\log x\right)}^{-n}dx}{x} \\ \mathrm{Let}\log x=t \\ \Rightarrow \frac{1}{x}dx=dt \\ \therefore I=\int t^{-n}.dt \\ =\frac{t^{-n+1}}{-n+1}+C \\ =\frac{{\left(\log x\right)}^{-n+1}}{-n+1}+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }t\mathit{=}\log \mathit{}x\right) \end{gathered}$$