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Question
\[\int\frac{e^x}{x}\left\{ \text{ x }\left( \log x \right)^2 + 2 \log x \right\} dx\]
Sum
Solution
\[\text{ Let I }= \int\frac{e^x}{x}\left[ x \left( \log x \right)^2 + 2\log x \right]dx\]
\[ = \int e^x \left[ \left( \log x \right)^2 + \frac{2\log x}{x} \right]dx\]
\[Here, f(x) = \left( \log x \right)^2 \]
\[ \Rightarrow f'(x) = \frac{2\log x}{x}\]
\[\text{ put e}^x f(x) = t\]
\[ \Rightarrow e^x \left( \log x \right)^2 = t\]
\[\text{ Diff both sides w . r . t x }\]
\[\left[ e^x \left( \log x \right)^2 + e^x \frac{2\log x}{x} \right]dx = dt\]
\[ \therefore I = \int dt\]
\[ = t + C\]
\[ = e^x \left( \log x \right)^2 + C\]
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