Evaluate $\int_{0}^{\pi/4}x\ln^{2}(\sin(x))dx$

Question

Here is a some what challenging log sine integral.

$$I=\int_{0}^{\pi/4}x\ln^{2}(\sin(x))dx$$

The upper limit of integration is $\frac{\pi}{4}$ instead of the usual $\frac{\pi}{2}$.

I tried the famous identity $\displaystyle\ln(\sin(x))=-\sum_{k=1}^{\infty}\frac{\cos(2kx)}{k}-\ln(2)$

But, the log is squared and I do not think Parseval is applicable here. But, I may be wrong. By Parseval, I mean $\displaystyle\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\cos(kx)\cos(nx)}{kn}$.

I have tried other means. I made the sub $t=\sin(x)$. This led to $\displaystyle\int_{0}^{\frac{1}{\sqrt{2}}}\frac{\sin^{-1}(t)\ln^{2}(t)}{\sqrt{1-t^{2}}}dt$

Now, I thought perhaps the famous sum $\displaystyle\frac{\sin^{-1}(t)}{\sqrt{1-t^{2}}}=\sum_{k=1}^{\infty}\frac{(2t)^{2k-1}}{k\binom{2k}{k}}$ could be used.

So, I integrated and got the form

$$I=\frac{\ln^{2}(2)}{16}\sum_{k=1}^{\infty}\frac{(k!)^{2}2^{k}}{k^{2}(2k)!}+\frac{\ln(2)}{8}\sum_{k=1}^{\infty}\frac{(k!)^{2}2^{k}}{k^{3}(2k)!}+\frac{1}{8}\sum_{k=1}^{\infty}\frac{(k!)^{2}2^{k}}{k^{4}(2k)!}$$

These sums actually evaluate to the correct integral result, so it would appear they are equivalent to said integral. But evaluating them may be even more daunting. Though the first one from the left evaluates to $\frac{{\pi}^{2}}{8}$ and the center one evaluates to

$$\frac{\ln(2)}{8}\left(\pi G-\frac{35}{16}\zeta(3)+\frac{{\pi}^{2}}{8}\ln(2)\right)\;.$$

This center one looks to be related to the integral:

$\displaystyle8\ln(2)\int_{0}^{\frac{\pi}{4}}\ln(\sin(x))dx=\frac{35}{16}\ln(2)\zeta(3)-\pi\ln(2)G-\frac{{\pi}^{2}\ln^{2}(2)}{4}$, which I got by expanding the square of the log sine and integrating.

This just leaves the sum to the right. It is tougher and I can find no closed form.

Alas, If I could evaluate it then I think the integral would be solved.

I also thought...how about a closed form of some sort for

$$\int_{0}^{\frac{\pi}{4}}x\sin^{a}(x)dx$$

that could be differentiated w.r.t 'a' and applied?. I do not know of a closed form for this, though.

I got an encouraging looking solution by expanding, making the sub $t=x/2$, then solving for the integral on the far right (which is the integral in question):

$\displaystyle \int_{0}^{\frac{\pi}{2}}x\ln^{2}(2\sin(x/2))dx$ $=\displaystyle \ln^{2}(2)\int_{0}^{\frac{\pi}{2}}xdx+8\ln(2)\int_{0}^{\frac{\pi}{4}}x\ln(\sin(x))dx+\color{red}4\int_{0}^{\frac{\pi}{4}}x\ln^{2}(\sin(x))dx$

But, it did not check numerically. (Error has now been fixed. - Editor) Numerically, the integral in question should evaluate to $I = 0.263605....$

Then again, I may just have a computation error in all of that.

So, does anyone know of a method for evaluating this log-sine integral?.

EDIT: Here is what Mathematica gave for the solution. It may be too involved to be worth the effort.

$\displaystyle I=\frac{37{\pi}^{4}}{92160}-\frac{\pi}{4}\sum_{k=1}^{\infty}\frac{\sin(\pi k/4)}{2^{k/2}k^{3}}$ $\displaystyle +\frac{\pi}{8}\ln(2)G+\frac{25}{768}{\pi}^{2}\ln^{2}(2)+\frac{\ln^{4}(2)}{384}+\sum_{k=1}^{\infty}\frac{\cos(\pi k/4)}{2^{k/2}k^{4}}-\frac{35}{128}\ln(2)\zeta(3)$.

Sometimes with these math engines the solution can be simplified down. But, maybe not in this case. Those complex polylogs may pose a problem.

Answer 1

Previously I mentioned the three series I arrived at while trying to evaluate the log-sin integral in question. I have their closed forms via Mathematica. They add up to the required result. That is the best I could do.

The first one can be done rather easily using the classic arc sin sum identity.

$\displaystyle \frac{\ln^{2}(2)}{16}\sum_{k=1}^{\infty}\frac{(k!)^{2}2^{k}}{k^{2}(2k)!}=\frac{\ln^{2}(2)}{16}\cdot \frac{{\pi}^{2}}{8}$

The second one can be derived from the arcsin identity as well, though it is a wee bit tougher.

$\displaystyle\frac{\ln(2)}{8}\sum_{k=1}^{\infty}\frac{(k!)^{2}2^{k}}{k^{3}(2k)!}=\frac{\pi\ln(2)G}{8}-\frac{35}{128}\zeta(3)\ln(2)+\frac{{\pi}^{2}}{64}\ln(2)$

The third one is nastier.

$\displaystyle\frac{1}{8}\sum_{k=1}^{\infty}\frac{(k!)^{2}2^{k}}{k^{4}(2k)!}=\frac{7}{768}{\pi}^{2}\ln^{2}(2)+\frac{\ln^{4}(2)}{384}+\frac{37{\pi}^{4}}{92160}-\frac{\pi}{4}\sum_{k=1}^{\infty}\frac{\sin(\pi k/4)}{2^{k/2}k^{3}}+\sum_{k=1}^{\infty}\frac{\cos(k\pi/4)}{2^{k/2}k^{4}}$.

One then sums them up.

Answer 2

It's been a while since I posted this problem. Well, I didn't give up. I doubt if anyone cares at this point, but I found a way to evaluate integrals of this form. It is tedious to say the least.

$\frac{\pi}{3}$ being the upper limit of integration is about the easiest, but it is even kind of difficult to do. Not to mention, the $\pi/2$ and notoriously difficult $\pi/4$ case.

It uses $\ln(1-e^{ix})=\ln(2\sin(x/2))-\frac{i}{2}(\pi-x)$ and various polylog identities.

I managed to evaluate $\displaystyle \int_{0}^{\frac{\pi}{3}}x\left(\ln(2\sin(x/2))-\frac{i}{2}(\pi-x)\right)^{2}dx=\frac{-7{\pi}^{4}}{4860}-\frac{8{\pi} i\zeta(3)}{9}+2L_{4}(e^{\frac{\pi i}{3}})+\frac{2{\pi}^{2}i}{9}\sum_{n=1}^{\infty}\frac{\sin(\frac{k\pi}{3})}{k^{2}}$.

If need be, the Clausen sum can be written in a closed form involving derivatives of digamma and what not.

This checks numerically with Mathematica which I was happy to see finally worked out.

This is the EASY one. Other upper limits, such as $\frac{\pi}{2}, \;\ \frac{\pi}{4}$ are monstrosities.

It is related to $\displaystyle \int_{0}^{x}\frac{\ln^{a}(x)\ln^{b}(1-x)}{x}dx=i^{a}\int_{0}^{x}t^{a}\left(\ln(2\sin(t/2))-\frac{i}{2}(\pi-t)\right)^{b}dt$.

Of course, to evaluate $\displaystyle\int_{0}^{\frac{\pi}{3}}x\ln^{2}(2\sin(x/2))dx$, one could expand the square and solve for the intended integral. But, this can be rather laborious because of $\displaystyle\pi \int_{0}^{\frac{\pi}{3}}x\ln(2\sin(x/2))dx=\frac{2\pi\zeta(3)}{3}-\frac{{\pi}^{2}}{3}\sum_{k=1}^{\infty}\frac{\sin(\frac{k\pi}{3})}{k^{2}}$ and

$\displaystyle \int_{0}^{\frac{\pi}{3}}x^{2}\ln(2\sin(x/2))dx=\mathcal{R}\left[-\frac{2\pi\zeta(3)}{9}-\frac{{\pi}^{2}}{9}\sum_{k=1}^{\infty}\frac{\sin(\frac{\pi k}{3})}{k^{2}}-2iL_{4}(e^{\frac{\pi i}{3}})\right]$

Note: I do not know any nice closed forms for $L_{4}$ polylogs.

i.e $\displaystyle L_{3}(e^{\frac{\pi i}{3}})=\frac{\zeta(3)}{3}+\frac{5{\pi}^{3}i}{162}$ and

$\displaystyle L_{2}(e^{\frac{\pi i}{3}})=\frac{{\pi}^{2}}{36}+i\sum_{k=1}^{\infty}\frac{\sin(\frac{\pi k}{3})}{k^{2}}$

I am sorry. I do not know how this post got displayed before the one several weeks ago.

Answer 3

A solution by Cornel Ioan Valean

First, let's observe that $$\log ^2\left(\frac{1}{2}\sin(2x)\right)+\log ^2(\tan(x))=2 \log ^2(\sin(x))+2 \log ^2(\cos(x)), \ 0<x<\pi/2, \tag1$$ and if we multiply both side of $(1)$ by $x$ and integrate from $x=0$ to $x=\pi/4$, we get $$\int_0^{\pi/4}x\log ^2\left(\frac{1}{2} \sin(2x)\right)\textrm{d}x+\int_0^{\pi/4}x\log ^2(\tan(x))\textrm{d}x$$ $$=2 \int_0^{\pi/4}x\log ^2(\sin(x))\textrm{d}x+2\int_0^{\pi/4}x \log ^2(\cos(x)) \textrm{d}x. \tag2$$

Since $$\int_0^{\pi/4}x \log ^2\left(\frac{1}{2} \sin(2x)\right)\textrm{d}x=\frac{1}{4}\int_0^{\pi/2}x\log^2\left(\frac{1}{2} \sin(x)\right)\textrm{d}x$$ $$=\frac{\log^2(2)}{4}\int_0^{\pi/2}x\textrm{d}x-\displaystyle\frac{\log(2)}{2}\underbrace{\int_0^{\pi/2}x\log(\sin(x))\textrm{d}x}_{\displaystyle 7/16 \zeta(3)-3/4\log(2)\zeta(2)}+\frac{1}{4}\int_0^{\pi/2}x \log ^2(\sin(x))\textrm{d}x$$ and $$\int_0^{\pi/4} x\log^2(\cos(x)) \textrm{d}x= \int_{\pi/4}^{\pi/2} \left(\frac{\pi}{2}-x\right) \log^2(\sin(x)) \textrm{d}x$$ $$=\int_{0}^{\pi/2} \left(\frac{\pi}{2}-x\right) \log^2(\sin(x)) \textrm{d}x-\int_{0}^{\pi/4} \left(\frac{\pi}{2}-x\right) \log^2(\sin(x)) \textrm{d}x$$ $$=\frac{\pi}{2}\underbrace{\int_{0}^{\pi/2}\log ^2(\sin(x))\textrm{d}x}_{\displaystyle \pi^3/24+\pi/2 \log^2(2)}-\int_{0}^{\pi/2}x\log ^2(\sin(x))\textrm{d}x-\frac{\pi}{2}\int_{0}^{\pi/4}\log ^2(\sin(x))\textrm{d}x$$ $$+\int_{0}^{\pi/4}x\log ^2(\sin(x))\textrm{d}x,$$

based on $(2)$, we obtain that

$$\int_0^{\pi/4} x \log ^2(\sin(x)) \textrm{d}x$$ $$=-\frac{15 }{16}\zeta(4)-\frac{39}{64}\log^2(2)\zeta (2)-\frac{7}{128} \log(2)\zeta(3)+\frac{\pi}{4}\int_0^{\pi/4} \log ^2(\sin (x)) \textrm{d}x$$ $$+\frac{9}{16}\int_0^{\pi/2} x \log ^2(\sin (x)) \textrm{d}x+\frac{1}{4} \int_0^{\pi/4} x \log^2(\tan(x)) \textrm{d}x$$ $$=\frac{1}{8}\log (2)\pi G+\frac{5}{384}\log ^4(2)+\frac{5}{32}\log^2(2)\zeta(2)-\frac{35}{128}\log(2)\zeta(3)+\frac{95}{256}\zeta(4)$$ $$-\frac{\pi}{4}\Im\biggr\{\operatorname{Li}_3\left(\frac{1+i}{2}\right)\biggr\}+\frac{5 }{16}\operatorname{Li}_4\left(\frac{1}{2}\right),$$

which is the desired closed-form.

In the calculations we also used that

$$\int_0^{\pi/4} \log^2(\sin (x))\textrm{d}x=\frac{23}{384}\pi^3+\frac9{32}\pi\log^2(2)+\frac{1}{2}\log(2)G-\Im\biggr\{\operatorname{Li}_3\left(\frac{1+i}{2}\right)\biggr\},$$ which can be extracted from this answer, next

$$\int_0^{\pi/2} x \log ^2(\sin (x))\textrm{d}x=\operatorname{Li}_4\left(\frac{1}{2}\right)+\frac{1}{24}\log^4(2)+\frac{1}{2}\log^2(2)\zeta(2)-\frac{19}{32}\zeta(4),$$ which is proved here and here, and finally $$\int_0^{\pi/4} x \log^2(\tan(x)) \textrm{d}x=\int_0^1 \frac{\arctan(x)}{1+x^2} \log^2(x) \textrm{d}x$$ $$=\frac{151}{128}\zeta(4)+\frac{1}{4}\log^2(2)\zeta(2)-\frac{7}{8}\log(2)\zeta(3)-\frac{1}{24}\log^4(2)-\operatorname{Li}_4\left(\frac{1}{2}\right),$$

which can be extracted by using a key advanced series from this answer, beautifully derived with the help of a Fourier series from the book, (Almost) Impossible Integrals, Sums, and Series.

End of (beautiful) story

A note: the solution nicely and completely avoids the necessity of using contour integration.

Answer 4

It is $$ \log(\cos x)=-\log 2+ix-\sum^{\infty}_{k=1}\frac{(-1)^k}{k}e^{-2ikx} $$ Squaring we get $$ \log^2(\cos x)=\log^2 2-x^2-2x\sum^{\infty}_{k=1}\frac{(-1)^k}{k}\sin(2kx)+2\log 2 \sum^{\infty}_{k=1}\frac{(-1)^k}{k}\cos(2kx)+ $$ $$ +2\sum^{\infty}_{k=1}\frac{(-1)^k}{k}\cos(2kx)H_{k-1}, $$ since $$ \sum_{1\leq k,l\leq n,k+l=n}\frac{1}{kl}=\frac{2}{n}H_{n-1}, $$ where $H_n=1^{-1}+2^{-1}+\ldots+n^{-1}$. Hence $$ \int^{\pi/4}_{0}x\log^2(\cos x)dx=\frac{\pi^4}{1024}-\frac{\pi}{4}G\log 2+\frac{\pi^2}{32}\log^22+\frac{21}{64}\zeta(3)\log 2+S_1,\tag 1 $$ where $$ S_1=2\sum^{\infty}_{k=1}(-1)^k\frac{-2+2\cos(k\pi/2)+k\pi\sin(k\pi/2)}{8k^3}H_{k-1}. $$ Hence $$ S_1:=\frac{1}{2}\sum^{\infty}_{k=0}\frac{H_{2k}}{(2k+1)^3}-\frac{\pi}{4}\sum^{\infty}_{k=0}\frac{(-1)^kH_{2k}}{(2k+1)^2}-\frac{1}{8}\sum^{\infty}_{k=0}\frac{H_{4k+1}}{(2k+1)^3}.\tag 2 $$ In the same way $$ \log^2(\sin x)=-\frac{\pi^2}{4}+\pi x-x^2+\log^2 2+(\pi-2x)\sum^{\infty}_{k=1}\frac{\sin(2kx)}{k}+ $$ $$ +2\log 2\sum^{\infty}_{k=1}\frac{\cos(2kx)}{k}+2\sum^{\infty}_{k=1}\frac{\cos(2kx)}{k}H_{k-1} $$ Hence as with cosine $$ \int^{\pi/4}_{0}x\log^2(\sin x)dx=\frac{11\pi^4}{3072}+\frac{\pi}{4}G\log 2+\frac{\pi^2}{32}\log^2 2-\frac{35}{64}\zeta(3)\log 2+S_2,\tag 3 $$ where $$ S_2=-\frac{1}{2}\sum^{\infty}_{k=0}\frac{H_{2k}}{(2k+1)^3}+\frac{\pi}{4}\sum^{\infty}_{k=0}\frac{(-1)^kH_{2k}}{(2k+1)^2}-\frac{1}{8}\sum^{\infty}_{k=0}\frac{H_{4k+1}}{(2k+1)^3}.\tag 4 $$