Integral in greatest integer and absolute functions

Integral in greatest integer and absolute functions

$$ \tag{1} \int_{1}^{4} \ln [x]\,dx $$ now we are given this problem ,
what i did was to write function as $$\int_1^4 1\cdot\ln [x]\,dx $$ and
integral by parts yielded $[x]\ln [x]-[x]$ now we can enter limits to get
the integral , have i done it correctly or am i missing something ? $$
\tag{2} \int_0^\pi |\cos(x)-\sin(x)|\,dx $$ what i did was $$\int_0^\pi
|\cos(x)| -\int_0^\pi|\sin(x)|\,dx $$ now since $\cos(x) \lt 0$ in
$(-\pi/2,\pi)$ i wrote the integral $$\int_0^{\pi/2} \cos(x) \,
dx-\int_{\pi/2}^\pi \cos(x) \, dx-\int_0^\pi \sin(x)\,dx$$ did i made any
mistake ?