# How to evaluate I=\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dxI=\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx

Find the value of
$I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$

We have the information that
$J=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos x)\ \mathrm dx=\dfrac{(\pi\ln{2})^2}{8}-\dfrac{\pi^4}{192}$

Tools Needed

Tool Use

Numerically, $(7)$ matches the integral. I’m working on the last harmonic sum. Both numerical integration and $(7)$ yield $0.0778219793722938643380944$.

Mathematica Help

Thanks to Artes’ answer on Mathematica, I have verified that these agree to 100 places.