Integral of 1/sqrt((x^2)+1)

Question

I have tried u substitution, but am struggling. I have seen people use trig-substitution, but that really doesn't make sense for me. Please help.

Answer 1

∫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.