Question

Evaluate:\[\int\frac{\left( 1 + \log x \right)^2}{x} \text{   dx }\]

Answer 1

\[\text{ Let } \int\frac{\left( 1 + \log x \right)^2}{x} \text{ dx }\]
\[\text{ Putting  1} + \log x = t\]
\[ \Rightarrow \frac{1}{x} dx = dt\]
\[ \therefore I = \int t^2 \cdot dt\]
\[ = \frac{t^3}{3} + C\]
\[ = \frac{\left( 1 + \log x \right)^3}{3} + C \left[ \because t = \left( 1 + \log x \right) \right]\]