Being $C : t \to\mathbb R^2$ a curve parametrized by $r(t) = (x(t),y(t))$ where $a \le t\le b$, a scalar field $f:\mathbb R^2 \to\mathbb R $, line integral is defined as: $$\int_{C}fds = \int_{a}^bf(r(t))\left \|r'(t) \right \|dt $$ where s(t) = $\int_{0}^{t}\left \| r'(t) \right \|dt $ being an arc length function.
But when I started Green's theorem I found some corollary where $C_1$ being parametrized by $r:x\to(x,\phi_1(x))$ given a function $P:\mathbb R^2\to \mathbb R$ and similarly $a\le x\le b$, I found that $$ \int_{C_1^+}P(r(x))dx = \int_a^b P(x,\phi_1(x))dx$$
My question is: Why can we change ds by dx and why are we missing $\left \| r'(t) \right \|$ in the second part of the last equation. I don't know if I can be clearer.
There are two kinds of line integrals. You wrote down the integral with respect to arclength, but the more important sort of integral, which shows up in physics (work) and in Green's Theorem, Stokes's Theorem, etc., is an integral of the form $$\int_C \vec F\cdot d\vec r = \int_C P\,dx + Q\,dy.$$ You can express this as an arclength integral by introducing the unit tangent vector $\vec T$ of the curve $C$: $$\int_C \vec F\cdot d\vec r = \int_C \vec F\cdot\vec T\,ds.$$