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Sum of n Terms

If ∫ ex ln x ...

Question

If $\int e^{x} (ln x + \frac{1}{x^{2}}) d x = e^{x} \cdot f (x) + C,$ where $C$ is the constant of integration, then the value of $f^{'} (0.5)$ is equal to

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Solution

$I = \int e^{x} (ln x + \frac{1}{x^{2}}) d x = \int e^{x} [(ln x - \frac{1}{x}) + (\frac{1}{x} + \frac{1}{x^{2}})] d x = e^{x} (ln x - \frac{1}{x}) + C [∵ \frac{d}{d x} (ln x - \frac{1}{x}) = \frac{1}{x} + \frac{1}{x^{2}}] \Rightarrow f (x) = ln x - \frac{1}{x} \Rightarrow f^{'} (x) = \frac{1}{x} + \frac{1}{x^{2}} ∴ f^{'} (0.5) = f^{'} (\frac{1}{2}) = 2 + 4 = 6$

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