Answer
$$\ln \left( {\frac{{3 + \sqrt 5 }}{2}} \right)$$
Work Step by Step
$$\eqalign{
& \int_3^7 {\frac{1}{{\sqrt {{x^2} - 4} }}} dx \cr
& {\text{Use Theorem 5}}{\text{.20 }}\int {\frac{{du}}{{\sqrt {{u^2} - {a^2}} }} = } \ln \left( {u + \sqrt {{u^2} - {a^2}} } \right) + C \cr
& \int_3^7 {\frac{1}{{\sqrt {{x^2} - 4} }}} dx = \left[ {\ln \left( {x + \sqrt {{x^2} - 4} } \right)} \right]_3^7 \cr
& {\text{Evaluate}} \cr
& = \ln \left( {7 + \sqrt {{7^2} - 4} } \right) - \ln \left( {3 + \sqrt {{3^2} - 4} } \right) \cr
& {\text{Simplifying}} \cr
& = \ln \left( {7 + \sqrt {45} } \right) - \ln \left( {3 + \sqrt 5 } \right) \cr
& = \ln \left( {\frac{{7 + \sqrt {45} }}{{3 + \sqrt 5 }}} \right) \cr
& {\text{Rationalizing}} \cr
& = \ln \left( {\frac{{7 + 3\sqrt 5 }}{{3 + \sqrt 5 }} \times \frac{{3 - \sqrt 5 }}{{3 - \sqrt 5 }}} \right) \cr
& = \ln \left( {\frac{{21 - 7\sqrt 5 + 9\sqrt 5 - 15}}{{9 - 5}}} \right) \cr
& = \ln \left( {\frac{{6 + 2\sqrt 5 }}{4}} \right) \cr
& = \ln \left( {\frac{{3 + \sqrt 5 }}{2}} \right) \cr} $$
