Answer
$$F'\left( x \right) = \frac{{\cos x}}{x}$$
Work Step by Step
$$\eqalign{
& F\left( x \right) = \int_\pi ^{\ln x} {\cos {e^t}} dt \cr
& {\text{Differentiate both sides with respect to }}x \cr
& F'\left( x \right) = \frac{d}{{dx}}\left[ {\int_\pi ^{\ln x} {\cos {e^t}} dt} \right] \cr
& {\text{Use the second fundamental theorem of calculus }} \cr
& {\text{and the chain rule }}\left( {{\text{see page 284}}} \right) \cr
& F'\left( x \right) = \cos {e^{\ln x}}\frac{d}{{dx}}\left[ {\ln x} \right] \cr
& F'\left( x \right) = \cos {e^{\ln x}}\left( {\frac{1}{x}} \right) \cr
& {\text{Simplifying}} \cr
& F'\left( x \right) = \cos x\left( {\frac{1}{x}} \right) \cr
& F'\left( x \right) = \frac{{\cos x}}{x} \cr} $$
