How To Integrate sqrt[x^2-25]/x^2 by Trigonometric Substitution

 

\begin{array}{l} \smallint \frac{{\sqrt {{x^{\bf{2}}} - {\bf{25}}} }}{{{x^{\bf{2}}}}}{\rm{d}}x\\\\ Let \Rightarrow 1 \cdot x = 5\sec \theta \;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{d}}x = 5\sec \theta \tan \theta {\rm{d}}\theta \\ Change{\rm{ }}the\;expression\\ \;\;{x^2} - 25 = 25{\sec ^2}\theta - 25\;\;\;\;\;\\ \;\;{x^2} - 25 = 25\left( {{{\sec }^2}\theta - 1} \right)\;\;\;\;\\ \;\;{x^2} - 25 = 25{\tan ^2}\theta \;\;\;\\ \;\;\sqrt {{x^2} - 25} = 5\tan \theta \\ \smallint \frac{{5\tan \theta }}{{5\sec \theta }} \cdot 5\sec \theta \cdot \tan \theta {\rm{d}}\theta = 5\smallint ta{n^2}\theta {\rm{d}}\theta \\ 5\smallint {\sec ^2}\theta - 1{\rm{d}}\theta \\ \Rightarrow 5\left( {\tan \theta - \theta } \right) + c \end{array}

\begin{array}{l} \Rightarrow 5\left( {\begin{array}{*{20}{c}} {\frac{{\sqrt {{x^2} - 25} }}{5} - }&{{{\sec }^{ - 1}}\frac{x}{5}} \end{array}} \right) + c \end{array}