Question

\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]

Answer 1

\[\text{ Let I }= \int e^x \left( \log x + \frac{1}{x} \right)dx\]

\[\text{ Here}, f(x) = \log x\]

\[ \Rightarrow f'(x) = \frac{1}{x}\]

\[\text{ put }\ e^x f(x) = t\]

\[ \Rightarrow e^x \log x = t\]

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

\[ e^x \log x + e^x \frac{1}{x} = \frac{dt}{dx}\]

\[ \Rightarrow e^x \left( \log x + \frac{1}{x} \right)dx = dt\]

\[ \therefore \int e^x \left[ \log  x + \frac{1}{x} \right]dx = \int dt\]

\[ \Rightarrow I = t + C\]

\[ = e^x \log x + C\]