Simplify x sqrt43x2 by making the substitution x 2 sqrt3tan

Simplify x /sqrt(4+3x^2) by making the substitution x= (2/ sqrt(3))tan u , 2u 2

Solution

x / sqrt ( 4 + 3x^2 )

x = (2 / sqrt 3) tan u

(2 / sqrt 3) tan u / ( 4 + 3 (2 / sqrt 3) tan u )^2 )

2 / sqrt 3 tan u / sqrt ( 4 + 4 tan^2 u )

2 / sqrt 3 tan u / sqrt ( 4 ( 1+ tan^2 u )

2 / sqrt 3 tan u / 2 sec u

tan u = sin u / cos u

sec u = 1 / cos u

on simplifying we get

( 2 / sqrt 3 ) sin u