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

Published On :2020-12-06 20:40:00

Post Image

 

begin{array}{l} smallint frac{{bf{1}}}{{xsqrt {{x^{bf{2}}} + {bf{4}}} }};dx\\ Let Rightarrow 1 cdot x = 2tan 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} = 4left( {{{tan }^2}theta + 1} right);;;;;\ ;;;;4 + {x^2} = 4{sec ^2}theta ;;;;\ ;;;;sqrt {4 + {x^2}} = 2sec theta \ smallint frac{{2{{sec }^2}theta }}{{2tan theta cdot 2sec theta }}{rm{d}}theta = frac{1}{2}smallint frac{{sectheta }}{{tantheta }}{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}{{sintheta }}{rm{d}}theta \ frac{1}{2}smallint csc theta dtheta \ Rightarrow - frac{1}{2}lnleft| {csctheta + cottheta } 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}