Resembling Resemblence

Calculus Level 5

If

\[I=\displaystyle\int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\dfrac{A}{{B}}+\dfrac{C}{D}\ln 2\pi\]

and \(\text{g.c.d}(A,B)=1\) also \(\text{gcd}(C,D)=1\) find \(A+B+C+D-1\)


Also try Part-2 and Part-3

Where, \(\Gamma(t)=\displaystyle\int_{0}^{\infty}x^{t-1}e^{-x}dx\)

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