Question

Write a value of $\int \frac{\log x^n}{x}dx$.

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \left(\frac{\log x^n}{x}\right)dx \\ =\int \frac{n\log x}{x}dx\left(\because \log x^a=a\log x\right) \\ \mathrm{Let}\log x=t \\ \Rightarrow \frac{1}{x}dx=dt \\ \therefore I=n\int tdt \\ =n.\frac{t^2}{2}+C \\ =\frac{n.{\left(\log \mathit{}x\right)}^2}{2}+C\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left(\mathit{\because }t=\log \mathit{}x\right) \end{gathered}$$