# 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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