Answer
$$\frac{7}{{24}}$$
Work Step by Step
$$\eqalign{
& \int_{\ln 2}^{\ln 3} {\frac{{{e^x} + {e^{ - x}}}}{{{e^{2x}} - 2 + {e^{ - 2x}}}}} \cr
& {\text{factoring the denominator}} \cr
& \int_{\ln 2}^{\ln 3} {\frac{{{e^x} + {e^{ - x}}}}{{{{\left( {{e^x} - {e^{ - x}}} \right)}^2}}}} dx \cr
& {\text{substitute }}u = {e^x} - {e^{ - x}},{\text{ }}du = \left( {{e^x} + {e^{ - x}}} \right)dx \cr
& {\text{express the limits in terms of }}u \cr
& x = \ln 2{\text{ implies }}u = {e^{\ln 2}} - {e^{ - \ln 2}} = 3/2 \cr
& x = \ln 3{\text{ implies }}u = {e^{\ln 3}} - {e^{ - \ln 3}} = 8/3 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_{\ln 2}^{\ln 3} {\frac{{{e^x} + {e^{ - x}}}}{{{{\left( {{e^x} - {e^{ - x}}} \right)}^2}}}} dx = \int_{3/2}^{8/3} {\frac{1}{{{u^2}}}} du \cr
& {\text{find the antiderivative}} \cr
& = \left. { - \frac{1}{u}} \right|_{3/2}^{8/3} \cr
& {\text{use the fundamental theorem}} \cr
& = - \left( {\frac{3}{8} - \frac{2}{3}} \right) \cr
& {\text{simplify}} \cr
& = \frac{7}{{24}} \cr} $$
