\int_0^{\pi/2}x\ln^2\left(2 \sin(x) \right)\,dx
Lets show the following result: Recall (see here ) Than Where we used the results: Proved here Proved here
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RECIPROCAL BINOMIAL SUM REPRESENTATION FOR ZETA(4)
Today´s post We will prove the following beautiful result To this end we will go through a lot of results ranging from Logsine integrals and Polylogarithms and Euler Sums. Let´s start by showing the connection between infinite sums involving the reciprocal of the binomial coefficients and Logsine integrals First recall the identity (proved here ) (1) Letting in (1) we immediately obtain (2) Now recall the following integral representation (see here a proof ) I want to show that (3) The following proof is based on Borwein (see references below) Proof: Then according to (3) we can write (4) To evaluate (4), we have to compute the integral on the R.H.S. of (4). As we did before , a good strategy is to use the following expansion for log squared: (5) proved here . Note in (5) the last term on the R.H.S.: An infinite series involving the Harmonic numbers and a trigonometric function. Unavoidably we will end up having to compute a nasty infinite ...