Question

\[\int e^x \left( \frac{x - 1}{2 x^2} \right) dx\]

Answer 1

\[Let I = \int e^x \left( \frac{x - 1}{2 x^2} \right)dx\]

\[ = \frac{1}{2}\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right)dx\]

\[here \frac{1}{x} = f(x) Put e^x f(x) = t\]

\[ \Rightarrow - \frac{1}{x^2} = f'(x)\]

\[let e^x \frac{1}{x} = t\]

\[Diff\ both\ sides\ w . r . t x\]

\[\left( e^x \frac{1}{x} + e^x \frac{- 1}{x^2} \right) = \frac{dt}{dx}\]

\[ \Rightarrow e^x \left( \frac{1}{x} - \frac{1}{x^2} \right)dx = dt\]

\[ \therefore I = \frac{1}{2}\int dt\]

\[ = \frac{t}{2} + C\]

\[ = \frac{e^x}{2x} + C\]