A beautiful function

Calculus Level 4

\[ \large \int_0^1 \ln ( \Gamma(x) ) \, dx \]

If the value of the integral above is equal to \[ \dfrac{ \ln(A \pi)}A \]

for some constant \(A\), find \(A\).

Notation: \( \Gamma(\cdot) \) denotes the gamma function.

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