int e^{x}left(frac{2}{x}-frac{2}{x^2}right) d | vedclass

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Question

(C) We know the standard integral formula: $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$. Given the integral: $I = \int e^{x} \left(\frac{2}{x} - \f

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$\frac{2 e^{x}}{x}+C$

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(C) We know the standard integral formula: $\int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$. Given the integral: $I = \int e^{x} \left(\frac{2}{x} - \frac{2}{x^2}\right) dx$. Let $f(x) = \frac{2}{x}$. Then, $f'(x) = \frac{d}{dx} (2x^{-1}) = -2x^{-2} = -\frac{2}{x^2}$. Substituting these into the formula: $I = \int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$. Therefore, $I = e^{x} \left(\frac{2}{x}\right) + C = \frac{2 e^{x}}{x} + C$.

$\int e^{x}\left(\frac{2}{x}-\frac{2}{x^2}\right) dx$ is equal to

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