Answer
$$\frac{\pi }{3}$$
Work Step by Step
$$\eqalign{
& \int_0^{\ln \left( {\sqrt 3 + 2} \right)} {\frac{{\cosh x}}{{\sqrt {4 - {{\sinh }^2}x} }}} dx \cr
& {\text{Let }}u = \sinh x,\,\,\,\,du = \cosh xdx \cr
& \int_0^{\ln \left( {\sqrt 3 + 2} \right)} {\frac{{\cosh x}}{{\sqrt {4 - {{\sinh }^2}x} }}} dx = \int_{\sinh \left( 0 \right)}^{\sinh \left[ {\ln \left( {\sqrt 3 + 2} \right)} \right]} {\frac{{du}}{{\sqrt {4 - {u^2}} }}} \cr
& = \int_0^{\sqrt 3 } {\frac{{du}}{{\sqrt {4 - {u^2}} }}} \cr
& {\text{Integrate}} \cr
& \int_0^{\sqrt 3 } {\frac{{du}}{{\sqrt {4 - {u^2}} }}} = \arcsin \left( {\frac{u}{2}} \right)_0^{\sqrt 3 } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \arcsin \left( {\frac{{\sqrt 3 }}{2}} \right) - \arcsin \left( 0 \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{\pi }{3} \cr} $$
