$$
\begin{align}
&\int_0^n\log(x)\,\mathrm{d}x+\int_0^1\log(\Gamma(x))\,\mathrm{d}x\\
&\int_0^1\left(\log(x)+\log(x+1)+\dots+\log(x+n-1)\vphantom{\int}\right)\,\mathrm{d}x
+\int_0^1\log(\Gamma(x))\,\mathrm{d}x\\
&=\int_0^1\log(\Gamma(x+n))\,\mathrm{d}x\\
&=\int_0^1\left(\log(\Gamma(n))+x\log(n)+O\left(\frac1n\right)\right)\,\mathrm{d}x\\
&=\log(\Gamma(n))+\frac12\log(n)+O\left(\frac1n\right)
\end{align}
$$
So
$$
n\log(n)-n+\int_0^1\log(\Gamma(x))\,\mathrm{d}x
=\color{#C00000}{n\log(n)-n+\frac12\log(2\pi/n)}+\frac12\log(n)+O\left(\frac1n\right)

For $0< x<1$, we have
$$
\begin{align}
\Gamma(x)\Gamma(1-x)
&=\int_0^\infty e^{-s}s^{x-1}\,\mathrm{d}s\int_0^\infty e^{-t}t^{-x}\,\mathrm{d}t\\
&=\int_0^\infty\int_0^\infty e^{-s-t}s^{x-1}t^{-x}\,\mathrm{d}s\,\mathrm{d}t\\
&=4\int_0^\infty\int_0^\infty e^{-s^2-t^2}s^{2x-1}t^{1-2x}\,\mathrm{d}s\,\mathrm{d}t\\
&=4\int_0^{\pi/2}\int_0^\infty e^{-r^2}\sin^{2x-1}(u)\cos^{1-2x}(u)r\,\mathrm{d}r\,\mathrm{d}u\\
&=2\int_0^{\pi/2}\tan^{2x-1}(u)\,\mathrm{d}u\\
&=2\int_0^\infty v^{2x-1}\,\mathrm{d}\arctan(v)\\