Let 2x=tan(y), dx=sec^2(y)dy/2.
√(4x^2+1)=√(tan^2(y)+1)=√sec^2(y)=sec(y).
The integrand becomes:
4sec^2(y)dy/2tan^2(y)sec(y)=2sec(y)dy/tan^2(y)=2cot(y)cosec(y)dy.
Differential of cosec(y)=differential of sin^-1(y)=
-sin^-2(y)cos(y)=-cot(y)cosec(y).
So ∫2cot(y)cosec(y)dy=-2cosec(y)+C=-√(4x^2+1)/x+C (where C is the constant of integration).
(This can also be written C-√(4+(1/x^2)). Differentiating we get -(1/2)(4+x^-2)^-(1/2)*(-2x^-3)=(1/(x^3√(4+x^-2)). In the denominator we take one x into the square root: 1/(x^2√(4x^2+1)). This is the original integrand, which confirms the integration solution.)