Answer
$$\frac{1}{3}\ln \left| {3x + \sqrt {9{x^2} - 100} } \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{\sqrt {9{x^2} - 100} }}} \cr
& u = 3x,{\text{ }}du = 3dx,{\text{ }}a = 10 \cr
& \int {\frac{{dx}}{{\sqrt {9{x^2} - 100} }}} = \frac{1}{3}\int {\frac{{du}}{{\sqrt {{u^2} - {a^2}} }}} \cr
& {\text{a matching integral in a table of integrals at the end of the book is the }} \cr
& {\text{formula }}69 \cr
& \int {\frac{{du}}{{\sqrt {{u^2} - {a^2}} }}} = \ln \left| {x + \sqrt {{u^2} - {a^2}} } \right| + C \cr
& \frac{1}{3}\int {\frac{{du}}{{\sqrt {{u^2} - {a^2}} }}} = \frac{1}{3}\ln \left| {x + \sqrt {{u^2} - {a^2}} } \right| + C \cr
& u = 3x,{\text{ }}a = 10 \cr
& = \frac{1}{3}\ln \left| {3x + \sqrt {9{x^2} - 100} } \right| + C \cr} $$
