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$\int \frac{\ln (e x^{x + 1}) + {(\ln (x^{\sqrt{x}}))}^{2}}{1 + (x \ln x) (\ln e^{2} x^{x})} d x = f (x) + c$ where c is integral constant and $f (1) = 0$ , then $e^{(e^{f (2)} - 1)}$ is equal to

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answer is B.

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Detailed Solution

$\int \frac{1 + (x + 1) \ln x + x {(\ln x)}^{2}}{1 + (x \ln x) (2 + x \ln x)} d x = \int \frac{1 + \ln x}{1 + x \ln x} d x = \ln (1 + x \ln x) + c ∴ e^{f (x)} = 1 + x \ln x e^{f (2)} - 1 = \ln 4$

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