Logarithms & Integrals

Calculus Level 5

I=0π2eln(cosxdcosxdx)dx \large I=\int^\frac \pi 2_0 e^{\ln\left ( \cos x \, \frac{\mathrm{d}\cos x}{\mathrm{d}x}\right ) } \mathrm{d}x

If complex logarithms and complex exponentials are not allowed, what is II equal to?

This question is derived from the Logarithm Definite Integral question.
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