How do I solve the integral $\int 2x\, g'(x^2+1)$?

Question

I would need some help verifying that I'm on the right track here.

Do I do it right by substituting $x^2+1 = u$ and $du = 2x\, dx$?

Would that mean that the following is correct: $\int g'(u)\, du = g(u) +c = g(x^2+1) + c$