Logarithms & Integrals

Calculus Level 5

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

This question is derived from the Logarithm Definite Integral question.
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