Question

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

Answer 1

\[\text{ Let I }= \int \frac{dx}{x \left( \log x \right)^n}\]
\[ = \int \frac{\left( \log x \right)^{- n} dx}{x}\]
\[\text{ Let log x } = t\]
\[ \Rightarrow \frac{1}{x}dx = dt\]
\[ \therefore I = \int t^{- n} . dt\]
\[ = \frac{t^{- n + 1}}{- n + 1} + C\]
\[ = \frac{\left( \log x \right)^{- n + 1}}{- n + 1} + C \left( \because t = \log x \right)\]