Question

Write a value of

\[\int\frac{\left( \log x \right)^n}{x} \text{ dx }\]

Answer 1

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