Question

\[\int\frac{\log x^2}{x} dx\]

Answer 1

\[\int\frac{\log x^2 dx}{x}\]
\[ = \int\frac{2 \log x}{x} dx\]
\[ = 2\int\frac{\log x}{x}dx\]
\[Let \log x = t\]
\[ \Rightarrow \frac{1}{x} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{1}{x} dx = dt\]
\[Now, 2\int\frac{\log x}{x}dx\]
\[ = 2\ ∫\text{ t  dt}\]
\[ = 2\left[ \frac{t^2}{2} \right] + C\]
\[ = t^2 + C\]
\[ = \left( \log x \right)^2 + C\]