Question

Write a value of

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

Answer 1

\[\text{ Let I }= \int \left( \frac{\log x^n}{x} \right) dx\]
\[ = \int \frac{n \log x}{x}dx \left( \because \log x^a = a \log x \right)\]
\[\text{ Let log x }= t\]
\[ \Rightarrow \frac{1}{x}dx = dt\]
\[ \therefore I = n \int \text{ t  dt}\]
\[ = n . \frac{t^2}{2} + C\]
\[ = \frac{n . \left( \log x \right)^2}{2} + C \left( \because t = \log x \right)\]