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Calculate $$\frac{\int_0^\pi x^3\ln(\sin x)dx}{\int_0^\pi x^2\ln(\sqrt{2}(\sin x)dx}$$

In this problem, I'm unable to understand how to start.

I tried applying integration by parts but I couldn't solve it. I also tried the various properties of definite integration but they were of no use. Maybe applying integration by parts (or DI method) successively may work but it leads to a form of $\frac{\infty}{\infty}$.

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    $\begingroup$ How in the denominator do you handle the logarithm's argument changing sign at $\pi/\sqrt{2}$? $\endgroup$ – J.G. May 20 at 16:11
  • $\begingroup$ Where did you get this integral? Is it a part of some larger problem? $\endgroup$ – Yuriy S May 21 at 13:49
  • $\begingroup$ No , It is from a hard book for JEE Advanced that my coaching school has given me. It is a standalone question , I have been asked that this equates to kπ , and we have to find the value of k , The answer provided is 1.5 , but no solution has been given. $\endgroup$ – RandomAspirant May 21 at 13:51
  • $\begingroup$ I highly doubt that either integral can be computed in closed form without knowledge of the Riemann zeta function at least. Perhaps you miscoppied the question, and wrote $$\ln(\sin(x\sqrt{2}))$$ instead of $$\ln(\sqrt{2}\sin x)$$ . I am not even sure if the integral on the denominator even exists. It would also help if you added more context to your question. $\endgroup$ – clathratus May 22 at 21:02
  • $\begingroup$ @Threesidedcoin I am so so so so so sorry. I'll make sure this doesn't happen in the future $\endgroup$ – RandomAspirant May 23 at 6:10
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We want to prove that: $$\frac{I}{J}=\frac{\int_0^\pi x^3\ln(\sin x)dx} {\int_0^\pi x^2\ln\left(\sqrt 2\sin x\right)dx}=\frac{3\pi}2$$

We can take the upper integral and perform Emperor's rule $(\pi-x\to x$ and sum a $0$ in the end): $$I=\int_0^\pi (\pi^3-3\pi^2x+3\pi x^2-x^3)\ln(\sin x)dx+ 3\pi(\underbrace{\ln \sqrt 2-\ln \sqrt 2}_{=0})\int_0^\pi x^2 dx$$ $$\small=\pi^3 \int_0^\pi \ln(\sin x)dx-3\pi^2 \int_0^\pi x\ln(\sin x)dx+3\pi\int_0^\pi x^2(\ln(\sin x)+\ln\sqrt 2)dx-I-{\pi^4}\ln \sqrt 2$$ $$\small 2I=\left(\pi^3-\frac{3\pi^3}{2}\right)\int_0^\pi \ln(\sin x)dx-{\pi^4}\ln \sqrt 2+3\pi\int_0^\pi x^2\ln(\sqrt2 \sin x)dx$$ $$\require{cancel} 2I=\cancel{\frac{\pi^3}{2}\cdot 2\pi \ln \sqrt 2}-\cancel{\pi^4 \ln \sqrt 2}+3\pi J\Rightarrow I=\frac{3\pi}2J$$


Things used above: $$K=\int_0^\pi x\ln(\sin x)dx=\int_0^\pi (\pi-x)\ln(\sin x)dx$$ $$2K=\int_0^\pi (x+\pi-x)\ln(\sin x)dx\Rightarrow K=\frac{\pi}{2}\int_0^\pi \ln(\sin x)dx$$ Split into two parts the latter integral in the point $\frac{\pi}{2}$ and reduce both bounds to $0\to \frac{\pi}{2}$. $$L=\int_0^\pi \ln(\sin x)dx=\int_0^\frac{\pi}{2} \ln(\sin x)dx+\int_0^\frac{\pi}{2} \ln(\cos x)dx$$ $$=\int_0^\pi \ln\left(\frac22\sin x\cos x\right)=\int_0^\frac{\pi}{2} \ln(\sin 2x)dx-\int_0^\frac{\pi}{2} \ln 2dx$$ $$=\frac12 \int_0^\pi \ln(\sin x) dx-\ln\sqrt 2 \int_0^{\pi} dx\Rightarrow L=-2\pi \ln\sqrt 2$$

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    $\begingroup$ Thanks a lot mate. It was really helpful $\endgroup$ – RandomAspirant May 23 at 9:41
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For those interest in the overkill approach, I will be providing closed forms for each integral with the use of special functions just for the hell of it.


We define $$p=\int_0^\pi x^3\ln\sin x\,dx$$ We recall the definition of the Clausen function of order $2$: $$\mathrm{Cl}_2(x)=-\int_0^x \ln\left|2\sin\frac{t}2\right|\,dt=\sum_{k\geq1}\frac{\sin kx}{k^2}$$ so $$-\ln\left(2\sin \frac{x}2\right)=\mathrm{Cl}_1(x)=\sum_{k\geq1}\frac{\cos kx}{k}$$ and thus $$\ln\sin x=-\ln2-\sum_{k\geq1}\frac{\cos2kx}{k}$$ then $$\begin{align} p&=-\int_0^\pi x^3\left(\ln2+\sum_{k\geq1}\frac{\cos2kx}{k}\right)dx\\ &=-\frac{\pi^4}4\ln2-\frac1{16}\sum_{k\geq1}\frac1{k^5}\int_0^{2k\pi}x^3\cos x\,dx \end{align}$$ We can use IBP to show that $$\int_0^{2k\pi}x^3\cos x\,dx=12\pi^2k^2$$ Which I leave to you as a challenge.

Long story short, $$p=-\frac{\pi^4}{4}\ln2-\frac{3\pi^2}4\zeta(3)$$ Where $\zeta(3)=\sum_{k\geq1}k^{-3}$ is Apery's Constant. And $\zeta(s)=\sum_{k\geq1}k^{-s}$ is the Riemann Zeta function.


Next up: $$q=\int_0^\pi x^2\ln(\sqrt{2}\sin x)\,dx=\frac{\pi^3}{6}\ln2+\int_0^\pi x^2\ln\sin x\,dx$$ Using the same series as last time, $$\begin{align} \int_0^\pi x^2\ln\sin x\,dx&=-\frac{\pi^3}{3}\ln2-\frac18\sum_{k\geq1}\frac1{k^4}\int_0^{2k\pi}x^2\cos x\,dx \end{align}$$ IBP shows that $$\int_0^{2k\pi}x^2\cos x\,dx=4\pi k$$ So of course $$\int_0^\pi x^2\ln\sin x\,dx=-\frac{\pi^3}{3}\ln2-\frac\pi2\zeta(3)$$ Hence $$q=-\frac{\pi^3}{6}\ln2-\frac\pi2\zeta(3)$$


So the ratio in question is $$\frac{p}{q}=\frac{\frac{\pi^4}{4}\ln2+\frac{3\pi^2}4\zeta(3)}{\frac{\pi^3}{6}\ln2+\frac\pi2\zeta(3)}=\frac32\pi$$

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