Logarithms, Trigonometry, Integrals (Part 4)

Calculus Level 5

\[\displaystyle \int _{ 0 }^{ \pi /4 }{ \frac { \ln { \left( \sin { (x) } \right) } \ln { \left( \cos { (x) } \right) } }{ \sin { (x) } \cos { (x) } } dx } \]

Given that the integral above is equal to \( \dfrac {\zeta(a)}b \), where \(a\) and \(b\) are integers, find \(a+b\).

Notation

  • \( \displaystyle \zeta (s) = \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ s } } } \).
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