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Calculus Level 5

∫ 0 π / 3 x [ ln ⁡ ( 2 sin ⁡ ( x 2 ) ) ] 2 d x = A π B C \int _{ 0 }^{ \pi /3 }{ x{ \left[ \ln { \left( 2\sin { \left( \dfrac { x }{ 2 } \right) } \right) } \right] }^{ 2 } \, dx } =\dfrac { { A\pi }^{ B } }{ C }

This equation holds true for positive integers A , B , C A,B,C with A A and C C being coprime. Evaluate A + B + C A+B+C .


The answer is 6501.

Aditya Kumar by Aditya Kumar , 22, India

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1 solution

The logsine integral defined by : L m n ( ϕ ) = − ∫ 0 ϕ x n l o g m − n − 1 ( 2 s i n x 2 ) d x L_{m}^{n}(\phi) = -\int_{0}^{\phi} x^n log^{m-n-1}(2sin\frac{x}{2})dx

The integral is therefore L 4 1 ( π 3 ) = 17 6480 π 4 L_{4}^{1}(\frac{\pi}{3}) = \frac{17}{6480}\pi^4

So A + B + C = 6501 \boxed{A+B+C=6501}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

This is not a complete solution. Post it with proper steps.

Aditya Kumar - 5 years, 1 month ago

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All the necessary steps (and the answer) are here . The Mathematica implementation found there is really useful!

Mark Hennings - 5 years, 1 month ago

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