Integrate $u_{\xi\eta}$ with respect to $\xi$.

Question

I'm studying about the non-homogeneous wave equation $u_{tt}(x,t)-c^2u_{xx}(x,t)=F(x,t)$. After the change of variables $\xi=x+ct,~\eta=x-ct$, this takes the form $$u_{\xi\eta}\left(\dfrac{\xi+\eta}{2},\dfrac{\xi-\eta}{2c}\right)=\dfrac{-1}{4c^2}F\left(\dfrac{\xi+\eta}{2},\dfrac{\xi-\eta}{2c}\right),$$ I want to integrate this with respect to $\xi$, my problem is when integrating LHS of the previous equation. According to the book "A first course in partial differential equations with complex variables and transformation methods - Weinberger", when we integrate with respect to $\xi$ we get: $$u_\eta\left(\dfrac{\xi+\eta}{2},\dfrac{\xi-\eta}{2c}\right)=\left.u_\eta\left(\dfrac{\bar\xi+\eta}{2},\dfrac{\bar\xi-\eta}{2c}\right)\right\rvert_{\bar\xi=\eta}+\int_\eta^\xi u_{\xi\eta}\left(\dfrac{\xi+\eta}{2},\dfrac{\xi-\eta}{2c}\right)d\bar\xi.$$ But I don't understand the evaluation $\bar\xi=\eta$ (maybe something about $\xi=\eta$ when $t=0$?), nor the other integral limits. I feel it is also something in fashion of $\int_0^x f(s)ds$ to not confuse variables. However, I know that if you integrate, for example, $\int f_{xy}(x,y)dx=f_y(x,y)+h(y)$ where $h$ is a function that depends on $y$. But I can't connect this ideas.