Inspired by Ronak Agarwal

Calculus Level 5

\[\displaystyle \int_0^{\frac{\pi}{2}} \bigg [ ( \cot{x} )\ \log{(\sin x)} \ \log^4{(\cos x)} \bigg ] \ dx \]

For positive integers \(a,b,c,d,e,f\), the above integral can be stated in the form of

\[ \frac{a}{b}\zeta^c {(d)} -\frac {\pi^e}{f} \]

Where \(a,b\) are coprime. Find \(f-e-d-c-b-a\).

Details and Assumptions

  • \(\zeta (x) \) denote the Riemann Zeta Function: \( \zeta (x) = \displaystyle \sum_{k=1}^\infty \frac 1 {k^x} \)

This problem is inspired by this

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