Question

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

Answer 1

\[\text{ Let I }= \int\frac{e^x}{x}\left[ x \left( \log x \right)^2 + 2\log x \right]dx\]

\[ = \int e^x \left[ \left( \log x \right)^2 + \frac{2\log x}{x} \right]dx\]

\[Here, f(x) = \left( \log x \right)^2 \]

\[ \Rightarrow f'(x) = \frac{2\log x}{x}\]

\[\text{ put  e}^x f(x) = t\]

\[ \Rightarrow e^x \left( \log x \right)^2 = t\]

\[\text{ Diff   both    sides   w . r . t x }\]

\[\left[ e^x \left( \log  x \right)^2 + e^x \frac{2\log x}{x} \right]dx = dt\]

\[ \therefore I = \int dt\]

\[ = t + C\]

\[ = e^x \left( \log x \right)^2 + C\]