Twist on log of sine and cosine integral $\int_{0}^{\frac{\pi}{2}}x\ln(\sin x)\ln(\cos x)dx$ 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.
 A: $$\zeta(4):=\sum_{n=1}^\infty\frac{1}{n^4}=\frac{\pi^4}{90}\Longrightarrow \zeta_2(4):=\sum_{n=1}^\infty\frac{1}{(2n)^4}=\frac{1}{16}\zeta(4)=\frac{\pi^4}{16\cdot 90}\Longrightarrow$$
$$\Longrightarrow\sum_{n=0}^\infty\frac{1}{(2n+1)^4}=\zeta(4)-\zeta_2(4)=\frac{15}{16}\frac{\pi^4}{90}=\frac{\pi^4}{96}$$
And you have your first question answered.
A: I thought I would come back and show what I done.  I am rather uneasy about this solution and you'll see why.
Using the identities mentioned previously:
$\displaystyle -\ln(\sin(x))=\sum_{k=1}^{\infty}\frac{\cos(2kx)}{k}+\ln(2)$
and $\displaystyle -\ln(\cos(x))=\sum_{k=1}^{\infty}\frac{(-1)^{k}\cos(2kx)}{k}+\ln(2)$
I subbed them in and arrived at:
$\displaystyle\int_{0}^{\frac{\pi}{2}}\left(\sum_{k=1}^{\infty}\frac{x\cos(2kx)}{k}+x\ln(2)\right)\left(\sum_{k=1}^{\infty}\frac{(-1)^{k}\cos(2kx)}{k}+\ln(2)\right)dx$
$=\displaystyle \int_{0}^{\frac{\pi}{2}}(\sum_{k=1}^{\infty}\frac{(-1)^{k}\cos(2kx)}{k}\sum_{k=1}^{\infty}\frac{x\cos(2kx)}{k} $ 
+$\displaystyle \ln(2)\sum_{k=1}^{\infty}\frac{x\cos(2kx)}{k}+\ln(2)\sum_{k=1}^{\infty}\frac{(-1)^{k}x\cos(2kx)}{k}+x\ln^{2}(2))dx$
Now, here I made an otherwise 'illegal' move.  I took the product of the sums under one summation.
$\displaystyle\int_{0}^{\frac{\pi}{2}}(\sum_{k=1}^{\infty}\frac{(-1)^{k}x\cos^{2}(2kx)}{k^{2}}$
$+\displaystyle\ln(2)\sum_{k=1}^{\infty}\frac{x\cos(2kx)}{k}$
$+\displaystyle\ln(2)\sum_{k=1}^{\infty}\frac{(-1)^{k}x\cos(2kx)}{k}$ $+x\ln^{2}(2))dx$
Switch the sum and integral:
$\displaystyle\sum_{k=1}^{\infty}(\underbrace{\int_{0}^{\frac{\pi}{2}}\frac{(-1)^{k}x\cos^{2}(2kx)}{k^{2}}dx}_{\text{[1]}} $ 
$+\displaystyle\underbrace{\ln(2)\int_{0}^{\frac{\pi}{2}}\frac{x\cos(2kx)}{k}dx}_{\text{[2]}}$ 
$+\displaystyle\underbrace{\ln(2)\int_{0}^{\frac{\pi}{2}}\frac{(-1)^{k}x\cos(2kx)}{k}dx}_{\text{[3]}}$ $+\underbrace{\ln^{2}(2)\int_{0}^{\frac{\pi}{2}}xdx}_{\text{[4]}})dx$
$[1]:\displaystyle  \frac{{\pi}^{2}}{16}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k^{2}}=\frac{-{\pi}^{4}}{192}$
$[2]:  \ln(2)\left(\frac{-1}{4}\sum_{k=1}^{\infty}\frac{1}{k^{3}}+\frac{1}{4}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k^{3}}\right)$
$[3]: \displaystyle \ln(2)\left(\frac{-1}{4}\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k^{3}}+\frac{1}{4}\sum_{k=1}^{\infty}\frac{1}{k^{3}}\right)$
$[4]: \displaystyle \ln^{2}(2)\int_{0}^{\frac{\pi}{2}}xdx=\frac{{\pi}^{2}\ln^{2}(2)}{8}$
[2] and [3] cancel one another out and I arrive at:
$\displaystyle\frac{{\pi}^{2}\ln^{2}(2)}{8}-\frac{{\pi}^{4}}{192}$
This worked out beautifully.    Is it a fluke or can one manipulate sums, like I done above, under certain conditions?.  Or did I actually manage to come up with a clever solution?. 
Also sorry for the undersized parentheses. Every time I tried enlarging them, the Latex would not display. I have been wrestling with this for sometime trying to get it all to display. Thanks All. 
A: Here is an approach that avoids using the Fourier series expansions for $\ln (\cos x)$ and $\ln (\sin x)$.
Consider
$$I = \int_0^{\pi/2} x^2 \ln (\sin x) \ln (\cos x) \, dx.$$
Enforcing a substitution of $x \mapsto \pi/2 - x$ gives
\begin{align*}
I &= \int_0^{\pi/2} \left (\frac{\pi}{2} - x \right )^2 \ln \left [\sin \left (\frac{\pi}{2} - x \right ) \right ] \ln \left [\cos \left (\frac{\pi}{2} - x \right ) \right ] \, dx\\
&= \int_0^{\pi/2} \left (\frac{\pi}{2} - x \right )^2 \ln (\cos x) \ln (\sin x) \, dx\\
&= \frac{\pi^2}{4} \int_0^{\pi/2} \ln (\sin x) \ln (\cos x) \, dx - \pi \int_0^{\pi/2} x \ln (\sin x) \ln (\cos x) \, dx\\
& \qquad + \int_0^{\pi/2} x^2 \ln (\sin x) \ln (\cos x) \, dx,
\end{align*}
or
$$\int_0^{\pi/2} x \ln (\sin x) \ln (\cos x) \, dx = \frac{\pi}{4} \int_0^{\pi/2} \ln (\sin x) \ln (\cos x) \, dx.$$
The integral appearing on the right can be found by differentiating the beta function. As
$$\text{B}(x,y) = 2 \int_0^{\pi/2} \cos^{2x - 1} \theta \sin^{2y - 1} \theta \, d\theta,$$
we see that
$$\int_0^{\pi/2} \ln (\sin \theta) \ln (\cos \theta) \, d\theta = \frac{1}{8} \partial_x \partial_y \text{B}(x,y) \Big{|}_{x=y=1/2},$$
giving
$$\int_0^{\pi/2} x\ln (\sin x) \ln (\cos x) \, dx = \frac{\pi}{32} \partial_x \partial_y \text{B}(x,y) \Big{|}_{x=y=1/2}.$$
Since
$$\partial_x \text{B}(x,y) = \text{B}(x,y) [\psi(x) - \psi (x + y)] \quad \text{and} \quad \partial_y \text{B}(x,y) = \text{B}(x,y) [\psi(y) - \psi (x + y)],$$
where $\psi (z)$ is the digamma function, we have
$$\partial_x \partial_y \text{B}(x,y) = \text{B}(x,y) \left [ \left \{\psi(x) - \psi (x + y) \right \} \left \{\psi(y) - \psi (x + y) \right \} - \psi^{(1)}(x + y) \right ].$$
Thus
\begin{align*}
\int_0^{\pi/2} x\ln (\sin x) \ln (\cos x) \, dx &= \frac{\pi}{32} \text{B} \left (\frac{1}{2}, \frac{1}{2} \right ) \left [ \left \{ \psi \left (\frac{1}{2} \right ) - \psi (1) \right \}^2 - \psi^{(1)}(1) \right ].
\end{align*}
Since
$$\text{B} \left (\frac{1}{2}, \frac{1}{2} \right ) = \frac{\Gamma (1/2) \Gamma (1/2)}{\Gamma (1)} = \pi,$$
and
$$\psi^{(1)}(1) = \zeta (2) = \frac{\pi^2}{6} \quad \text{and} \quad \psi \left (\frac{1}{2} \right ) - \psi (1) = - 2\ln (2),$$
we have
$$\int_0^{\pi/2} x\ln (\sin x) \ln (\cos x) \, dx = \frac{\pi^2}{8} \ln^2 (2) - \frac{\pi^4}{192},$$
as expected.
A: First, evaluate
\begin{align}
J=&\int_{0}^{\frac{\pi}{2}}{\ln^2(2\sin 2x)}\ dx\>\>\>\>\>\>\>(2x\to x)\\
=& \int_{0}^{\frac{\pi}{2}}\ln^2(2\sin x)\ dx
=\frac14 \int_{0}^{\frac\pi2}
[\ln^2(2\sin 2x)+\ln^2(\tan x)]\ dx\\
 =&\ \frac14 J + \frac14\int_{0}^{\frac\pi2}
\ln^2(\tan x)\ dx=\frac13 \int_{0}^{\frac\pi2}
\ln^2(\tan x)\ dx
\end{align}
Then, use $J$ to integrate
\begin{align}
 &\int_{0}^{\frac{\pi}{2}}{x\ln(2\sin x)\ln(2\cos x)}\ dx\>\>\>\>\>({x\to \frac\pi2-x})\\
=& \ \frac\pi4 \int_{0}^{\frac\pi2}\ln(2\sin x)\ln(2\cos x)\ dx\\
=&\ \frac\pi{16}\int_{0}^{\frac\pi2}
[\ln^2(2\sin 2x)-\ln^2(\tan x)]\ dx\\
=& -\frac\pi{24} \int_{0}^{\frac\pi2}
\ln^2(\tan x)dx
=-\frac{\pi}{24} \cdot \frac{{\pi}^{3}}{8}=-\frac{{\pi}^{4}}{192} \\
\end{align}
which yields
$$\int_{0}^{\frac{\pi}{2}} x\ln(\sin x)\ln(\cos x)\ dx
= \frac{{\pi}^{2}}{8} \ln^{2}2-\frac{{\pi}^{4}}{192}
$$
