Problem with integrating the differential equation more than once

[LagrangeEuler]

Starting from equation
$$\frac{dy}{dx}=\int^x_0 \varphi(t)dt$$
we can write
$$dy=dx\int^x_0 \varphi(t)dt$$
Now I can integrate it
$$\int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^x_0\varphi(t)dt$$
Is this correct?
Or I should write it as
$$\int^{y(x)}_{y(0)}dt=\int^x_0dx'\int^{x'}_0\varphi(t)dt$$
Best wishes in new year and thank you for the answer.

 

[dRic2]

$f(x) = \int_0^x \phi(t) dt$ so $y(x) = \int_0^x f(x') dx' + const$
Thus, it is the second