Dilogarithm integral of the form Li 2 ( sin 8 x ) \operatorname{Li}_2(\sin^8x)

Calculus Level 4

Evaluate 1 π 0 π 2 Li 2 ( sin 8 x ) d x \frac{1}{\pi}\int_0^{\frac{\pi}{2}}\operatorname{Li}_2\left(\sin^8 x\right)dx

The challenging work is to find the closed form for the original integral above .

Notation: Li 2 ( x ) \operatorname{Li}_2(x) is dilogarithm function.


The answer is 0.18433.

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1 solution

Naren Bhandari
Mar 18, 2021

The closed form for the integral is This result is acquired by the exploiting corollaries ( see page 8-9) of Theorem 3.2 of my article here at Researchgate.

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