"I am Back" says Integration Part 4

Calculus Level 5

I=0π/2log(cos(x)) log2(sin(x)) dxI=\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\ \log^ 2 (\sin(x)) \ \mathrm d x }

If 16I16I can be represented in the form of aπζ(b)cπlogd(f) a\pi \zeta(b) - c\pi\log^d (f)

Find a+b+c+d+fa+b+c+d+f

Details and Assumptions

  • a,b,c,d,fa,b,c,d,f are positive integers not neccasarily distinct.

  • ζ \zeta denote the Riemann Zeta Function.

  • log\log is the natural logarithm.

  • ff is not a multiple of a perfect power of any integer greater than 11.


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