Degrees of Definite Integral

Calculus Level 5

\[\large I= \int_0^\pi x^3\ln\sin x\, dx\\\large k = \int_0^\pi x^2\ln\left(\sqrt2\sin x\right)\,dx\]

Let the two integrals be defined as given above. If \(I\) can be written as \(I = \dfrac{A\pi^B}C k\), where \(A\), \(B\) and \(C\) are positive integers with \(A\) and \(C\) being coprime integers. Find \(A+B+C\).

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