Question

\[\int\frac{\log x}{x^n}\text{  dx }\]

Answer 1

`  ∫   1/x^n   log  x   dx `
`  " Taking  log x as the first function and "{1}/ {x^n}"  as the second function  " ` 
\[ = \log x\int\frac{1}{x^n}dx - \int\left( \frac{d}{dx}\log x\int\frac{1}{x^n}dx \right)dx\]
\[ = \log x\left( \frac{x^{- n + 1}}{- n + 1} \right) - \int\frac{1}{x}\left( \frac{x^{- n + 1}}{- n + 1} \right)dx\]
\[ = \log x\left( \frac{x^{- n + 1}}{- n + 1} \right) - \int\frac{x^{- n}}{- n + 1}dx\]
\[ = \log x\left( \frac{x^{- n + 1}}{- n + 1} \right) - \frac{x^{- n + 1}}{\left( - n + 1 \right)^2} + C\]
\[ = \log x\left( \frac{x^{1 - n}}{1 - n} \right) - \frac{x^{1 - n}}{\left( 1 - n \right)^2} + C\]