Answer
$$\ln \left| {\frac{{x + \sqrt {{x^2} - 49} }}{7}} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{\sqrt {{x^2} - 49} }}} \cr
& {\text{substitute }}x = 7\sec \theta ,{\text{ }}dx = 7\sec \theta \tan \theta d\theta \cr
& = \int {\frac{{7\sec \theta \tan \theta d\theta }}{{\sqrt {{{\left( {7\sec \theta } \right)}^2} - 49} }}} \cr
& = \int {\frac{{7\sec \theta \tan \theta d\theta }}{{\sqrt {49{{\sec }^2}\theta - 49} }}} \cr
& = \int {\frac{{7\sec \theta \tan \theta d\theta }}{{7\sqrt {{{\sec }^2}\theta - 1} }}} \cr
& = \int {\frac{{7\sec \theta \tan \theta d\theta }}{{7\sqrt {{{\tan }^2}\theta } }}} \cr
& = \int {\sec \theta d\theta } \cr
& {\text{integrating}} \cr
& = \ln \left| {\sec \theta + \tan \theta } \right| + C \cr
& {\text{substituting tan}}\theta {\text{ and sec}}\theta \cr
& = \ln \left| {\frac{x}{7} + \frac{{\sqrt {{x^2} - 49} }}{7}} \right| + C \cr
& {\text{simplify}} \cr
& = \ln \left| {\frac{{x + \sqrt {{x^2} - 49} }}{7}} \right| + C \cr} $$
