∫dx/√(x²+1).
Let x=tan(y), x²+1=sec²(y), dx=sec²(y)dy=(x²+1)dy.
∫(x²+1)dy/√(x²+1)=∫√(x²+1)dy=∫sec(y)dy.
Multiply integrand by [(sec(y)+tan(y))/(sec(y)+tan(y))]:
∫[(sec²(y)+sec(y)tan(y))/(tan(y)+sec(y))]dy=
ln|tan(y)+sec(y)|+C, where C is a constant.
So we have ln(x+√(x²+1))+C or ln[a(x+√(x²+1))], where a=e^C.