∫ √(x^2+4)/x^2dx
let
x=2tany
dx = 2(secy)^2 dy
∫ √(x^2+4)/x^2dx
=(1/2)∫ [secy/(tany)^2]dy
=(1/2)∫ [cosy/(siny)^2]dy
= -(1/2)(1/siny) + C
=-(1/2)[√(x^2+4) /x] + C
∫ x/sin(x^2)dx
=(1/2)∫ csc(x^2) dx^2
=(1/2)ln|csc(x^2)-cot(x^2)| + C