I'm trying to understand when it is intuitively obvious that the ladder method would be best used to tackle a problem in quantum mechanics.
Evaluate:
$$\frac{\int _0 ^\pi x^3 \ln (\sin x) \mathrm d x}{\int _0 ^\pi x^2 \ln(\sqrt 2 \sin x) \mathrm d x}$$
I tried expressing the integral in the numerator as a multiple of the integral in the denominator. To do this, I used the following property of definite integrals:
\begin{alig...
$$\begin{align}\log(\frac{\sin x}{x})
= & \log\prod_{n=1}^{\infty}\left( 1 - \frac{x^2}{n^2\pi^2}\right)
= \sum_{n=1}^{\infty}\log\left(1 - \frac{x^2}{n^2\pi^2}\right)\\
\stackrel{\color{blue}{^{[1]}}}{=} & -\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{1}{k}\left(\frac{x}{n\pi}\right)^{2k}
\stack...
We want to prove that: $$\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$$
Let's take the upper integral and substitute $\pi-x\to x$ and add a $0$ term in the end:
$$\Rightarrow I=\int_0^\pi (\pi^3-3\pi^2x+3\pi x^2-x^3)\ln(\sin x)dx+ 3\...
