Question

∫√arctg2x/(1+4x^2) dx

Answer 1

arctg2x=t
dt=2dx/(1+4x²)
$\int\limits { \sqrt{arctg2x} }/(1+4x^2) \, dx =1/2 \int\limits { \sqrt{t} } \, dt =1/2*2/3 \sqrt{arctg^32x} +C=$$1/3* \sqrt{arctg^32x} +C$