I'm pretty shaky with DEs myself, and the notation always confuses the heck out of me, but nobody else has answered this yet, so I thought I'd take a stab at it. Hopefully my answer will at least give you something to work with (or encourage better answers):
The $\int_{x_0}^{x} M(s, y_0) \, ds$ is probably just changing the name of the first variable inside $M$'s parenthesis. It is used to get get the result in the correct letter, $x$, after integrating.
Note also that the book probably uses the notation that $x$ is a variable, and $x_0$ is a fixed unknown (a.k.a. constant).
Now let me change notations here for a minute. Have you seen where sometimes they represent a function with a lowercase letter and the function's integral with an uppercase letter (e.g., $M'(x, y) = m(x, y)$)? I'm going to use that notation and try to find $\int_{x_0}^{x} m(s, y_0) \, ds$.
If the integral of $m(s, y_0)$ is $M(s, y_0)$ (that is, $\int m(s, y_0) \, ds = M(s, y_0)$), then
$\int_{x_0}^{x} m(s, y_0) \, ds = \left. M(s, y_0) \right|_{x_0}^{x} = M(x, y_0) - M(x_0, y_0)$
Since the last term, $M(x_0, y_0)$, is a number (aka constant), taking the derivative of this with respect to (wrt) $x$ yields $m(x, y_0)$.
Next, in the textbook's answer they used the definite integral of $N$ to get the value for $\psi$. That is if the partial derivative of $\psi$ wrt $y$ equals $N$ (e.g. $\frac{\partial \psi}{\partial y} = N$), then
$\psi(x, y) = \int_{y_0}^{y} \frac{\partial \psi}{\partial y} \, dy = \int_{y_0}^{y} N(x, t) \, dt = \int N(x, y) \, dy - \int N(x, y_0) \, dy$
The first term on the right hand side (rhs), $\int N(x, y) \, dy$, is written that way because we don't know the integral of $N$ (it's the capital of capital $n$, or a really big $N$ :/ ). It's not an indefinite integral in the usual sense, it's just a notation to indicate "the function that you'd get when you integrate $N(x, t)$ with respect to $t$ and then evaluate that function at $t = y$".
The second term, $\int N(x, y_0) \, dy$, is obtained the same way and it means "the function you get when you integrate $N(x, t)$ wrt $t$ and then evaluate the function at $t = y_0$". It is a function of $x$, and I think they just renamed it $h$, so that $h(x) = \int N(x, y_0)$.
Next, they took the partial derivative of $\psi$ wrt $x$ and set it equal to $M$ and solved for h'(x). I don't understand why they reversed the integral and partial derivative signs, but I guess that's allowed, and I don't understand the rest of the book's answer from this point on. But I hope I at least clarified why the answer looked like it was using indefinite integrals. Maybe someone can now chip in with an explanation of the rest of the text's answer.