Integrate the function: $e^{x}\left(\frac{1}{x}-\frac{1}{x^{2}}\right)$
(A) Let $I = \int e^{x} \left[ \frac{1}{x} - \frac{1}{x^{2}} \right] dx$.
We know the standard integral formula: $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$.
Here,let $f(x) = \frac{1}{x}$.
Then,the derivative is $f'(x) = -\frac{1}{x^{2}}$.
Substituting these into the formula,we get:
$I = e^{x} \left( \frac{1}{x} \right) + C = \frac{e^{x}}{x} + C$,where $C$ is an arbitrary constant.
We know the standard integral formula: $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$.
Here,let $f(x) = \frac{1}{x}$.
Then,the derivative is $f'(x) = -\frac{1}{x^{2}}$.
Substituting these into the formula,we get:
$I = e^{x} \left( \frac{1}{x} \right) + C = \frac{e^{x}}{x} + C$,where $C$ is an arbitrary constant.
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