I Problem with integrating the differential equation more than once

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