"I am Back" says Integration Part 4

Calculus Level 5

\[I=\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\ \log^ 2 (\sin(x)) \ \mathrm d x } \]

If \(16I\) can be represented in the form of \( a\pi \zeta(b) - c\pi\log^d (f) \)

Find \(a+b+c+d+f\)

Details and Assumptions

  • \(a,b,c,d,f\) are positive integers not neccasarily distinct.

  • \( \zeta \) denote the Riemann Zeta Function.

  • \(\log\) is the natural logarithm.

  • \(f\) is not a multiple of a perfect power of any integer greater than \(1\).

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