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how can we compute the integral of PDE if both the PDE and integral is with respect to $x$?

For example:

$int w(x) , partial_{x} , [mu(x),partial_{x},u(x,t)]dx$

I know we have to do integration by parts, but how do we do integration of:

$int partial_{x} , [mu(x),partial_{x},u(x,t)]dx$

The book says that the result is:

$int mu(x) , partial_{x} ,w(x),partial_{x},u(x,t)dx$

I cannot get that result.

Thanks

You are integrating only in $x$. As such you should treat $y$ constant. Let
$y$ be fixed, and set $U(x) := u(x,y)$. Then we have $partial_x u = U'$, and the integral is
$$ int_0^L w (mu u')' dx$$

Integration by parts:
$$ int_0^L w (mu U')' dx = w mu U'Big|_0^L - int_0^L w' (mu U') dx$$
the boundary term disappears using the Neumann boundary condition.

PS note that $w:Gtomathbb R$ is a one-variable function so some people would say you cannot write $partial_x w$, only $w'$. The person who wrote your notes is not this kind of person, so you should get used to treating $partial_x$ as a synonym for $d/dx$ if you want to continue reading.