I ran across this integral and have not been able to evaluate it.
$\displaystyle \int_{0}^{\frac{\pi}{2}}x\ln(\sin(x))\ln(\cos(x))dx=\frac{{\pi}^{2}\ln^{2}(2)}{8}-\frac{{\pi}^{4}}{192}$
I had some ideas. Perhaps some how arrive at $\displaystyle\frac{1}{2}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{4}}=\frac{{\pi}^{4}}{192}$.
and $\displaystyle \ln(2)\int_{0}^{\frac{\pi}{2}}x\ln(2)dx=\frac{{\pi}^{2}\ln^{2}(2)}{8}$
by using the identity $\displaystyle\sum_{k=1}^{\infty}\frac{x\cos(2kx)}{k}=-x\ln(\sin(x))-x\ln(2)$
and/or $\displaystyle \ln(\cos(x))=-\ln(2)-\sum_{k=1}^{\infty}\frac{(-1)^{k}\cos(2kx)}{k}$
I have used the first one to evaluate $\displaystyle\int_{0}^{\frac{\pi}{2}}x\ln(\sin(x))dx$, so I thought perhaps it could be used in some manner here.
I see some familiar things in the solution, but how to get there?.
Does anyone have any clever ideas?.
Thanks.
Best Answer
$$\zeta(4):=\sum_{n=1}^\infty\frac{1}{n^4}=\frac{\pi^4}{90}\Longrightarrow \zeta_2(4):=\sum_{n=1}^\infty\frac{1}{(2n)^4}=\frac{1}{16}\zeta(4)=\frac{\pi^4}{16\cdot 90}\Longrightarrow$$
$$\Longrightarrow\sum_{n=0}^\infty\frac{1}{(2n+1)^4}=\zeta(4)-\zeta_2(4)=\frac{15}{16}\frac{\pi^4}{90}=\frac{\pi^4}{96}$$
And you have your first question answered.