When integrals meet abominable numbers

Calculus Level 5

\[ \int_0^{\frac \pi 2 } \bigg [ ( \sin^{12} x ) \ln (\sin x) \bigg ] \ dx = \frac {\pi}{A \cdot 2^B} \ (C - D \ln 2) \]

The equation above is true for integer constants \(A,B,C,D \). Find sum of the sum of digits of \(A,B,C\) and \(D\)


\(A\), \(B\) are \(2\)-digit numbers which differ by \(1\). \(C\) and \(D\) are 5 digit numbers of which \(C\) has only two prime factors.


This question may be extremely difficult/calculative if done in certain ways. But a general method may be devised.... NOT for the light hearted.


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