How to Integrate 1/xsqrt[x^2+4] by Trigonometric Substitution

 

\begin{array}{l} \smallint \frac{{\bf{1}}}{{x\sqrt {{x^{\bf{2}}} + {\bf{4}}} }}\;dx\\\\ Let \Rightarrow 1 \cdot x = 2\tan \theta \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{d}}x = 2{\sec ^2}\theta {\rm{d}}\theta \\ Change{\rm{ }}the\;expression\\ \;\;\;\;\;4 + {{\rm{x}}^2} = 4ta{n^2}\theta + 4\;\;\;\;\;\\ \;\;\;\;4 + {x^2} = 4\left( {{{\tan }^2}\theta + 1} \right)\;\;\;\;\;\\ \;\;\;\;4 + {x^2} = 4{\sec ^2}\theta \;\;\;\;\\ \;\;\;\;\sqrt {4 + {x^2}} = 2\sec \theta \\ \smallint \frac{{2{{\sec }^2}\theta }}{{2\tan \theta \cdot 2\sec \theta }}{\rm{d}}\theta = \frac{1}{2}\smallint \frac{{sec\theta }}{{tan\theta }}{\rm{d}}\theta \\ \frac{1}{2}\smallint \frac{1}{{\cos \theta }} \cdot \frac{{\cos \theta }}{{\sin \theta }}{\rm{d}}\theta = \frac{1}{2}\smallint \frac{1}{{sin\theta }}{\rm{d}}\theta \\ \frac{1}{2}\smallint \csc \theta d\theta \\ \Rightarrow - \frac{1}{2}ln\left| {csc\theta + cot\theta } \right| + c \end{array}

\begin{array}{l} \Rightarrow - \frac{1}{2}\ln \left| {\frac{{\sqrt {4 + {x^2}} }}{x} + \frac{2}{x}} \right| + c \end{array}