let u = x^1/2, u^2 = x and then 2u du = dx.
Rewriting: 2*integral (u/u+1) du = 2*integral [u+1-1]/(u+1)du
= 2[integral (u+1)/(u+1)du - integral 1/(u+1)du]
= 2[integral du - integral 1/(u+1)du] Let v = u+1, dv=du
= 2[u - integral dv/v = 2u - ln|v|] + C
= 2[x^1/2 - ln|u+1|] + C = 2[x^1/2 - ln|(x^1/2 + 1)|] + C