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$\Large{ I=\int^{\Large{\frac{\pi}{2}}}_{0} \left[e^{\ln\left ( \cos x \, \frac{\mathrm{d}(\cos x)}{\mathrm{d}x}\right ) }\right] \, \mathrm{d}x}$ If complex logarithms & complex exponentials are NOT allowed, then $I$ is equal to

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