Question

$\int \frac{\log x}{x^n}dx$

Answer 1

$$\begin{gathered} \int \frac{1}{x^n}\log xdx \\ \mathrm{Taking}\log x\mathrm{as}\mathrm{the}\mathrm{first}\mathrm{function}\mathrm{and}\frac{1}{x^n}\mathrm{as}\mathrm{the}\mathrm{second}\mathrm{function}. \\ =\log x\int \frac{1}{x^n}dx-\int \left(\frac{d}{dx}\log x\int \frac{1}{x^n}dx\right)dx \\ =\log x\left(\frac{x^{-n+1}}{-n+1}\right)-\int \frac{1}{x}\left(\frac{x^{-n+1}}{-n+1}\right)dx \\ =\log x\left(\frac{x^{-n+1}}{-n+1}\right)-\int \frac{x^{-n}}{-n+1}dx \\ =\log x\left(\frac{x^{-n+1}}{-n+1}\right)-\frac{x^{-n+1}}{{\left(-n+1\right)}^2}+C \\ =\log x\left(\frac{x^{1-n}}{1-n}\right)-\frac{x^{1-n}}{{\left(1-n\right)}^2}+C \end{gathered}$$