How To Integrate sqrt[x^2-25]/x^2 by Trigonometric Substitution
Published On :2021-01-28 20:46:00
begin{array}{l}
smallint frac{{sqrt {{x^{bf{2}}} - {bf{25}}} }}{{{x^{bf{2}}}}}{rm{d}}x\\
Let Rightarrow 1 cdot x = 5sec theta ;;;;;;;;;;;;\
;;;;;;;;;;;;;{rm{d}}x = 5sec theta tan theta {rm{d}}theta \
Change{rm{ }}the;expression\
;;{x^2} - 25 = 25{sec ^2}theta - 25;;;;;\
;;{x^2} - 25 = 25left( {{{sec }^2}theta - 1} right);;;;\
;;{x^2} - 25 = 25{tan ^2}theta ;;;\
;;sqrt {{x^2} - 25} = 5tan theta \
smallint frac{{5tan theta }}{{5sec theta }} cdot 5sec theta cdot tan theta {rm{d}}theta = 5smallint ta{n^2}theta {rm{d}}theta \
5smallint {sec ^2}theta - 1{rm{d}}theta \
Rightarrow 5left( {tan theta - theta } right) + c
end{array}
begin{array}{l}
Rightarrow 5left( {begin{array}{*{20}{c}}
{frac{{sqrt {{x^2} - 25} }}{5} - }&{{{sec }^{ - 1}}frac{x}{5}}
end{array}} right) + c
end{array}
