Conjecture <msubsup> &#x222B;<!-- ∫ --> 0 1 </msubsup> ln

Sattelhofsk

Sattelhofsk

Answered question

2022-06-14

Conjecture 0 1 ln ( ln 2 x + arccos 2 x ) 1 x 2 d x = ? π ln ln 2

Answer & Explanation

Sydnee Villegas

Sydnee Villegas

Beginner2022-06-15Added 22 answers

Substitute x cos x to obtain an equivalent formulation:
0 π 2 log ( x 2 + log 2 cos x ) d x = π log log 2.
boloman0z

boloman0z

Beginner2022-06-16Added 10 answers

Hint: Note that
ln 2 cos t + t 2 = ln 2 cos t ln 2 e i t = ( ln e i t cos t ) ( ln e i t cos t )
Then let e 2 i t = s to obtain a contour integral surrounding the simple pole at s = 0

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