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