Análisis

Question

En un problema de la teoría de la dispersión, surge esta integral: $$\displaystyle{\int\limits_0^{\pi/2} \frac{t \ln (1-\sin{t})}{\sin t} dt}$$ I have tried a number of approaches to evaluating the integral, which I suspect has a closed form solution. The reason is that I generated a numerical value for the integral, $ - 3.87578458503 \ ldots$ and after a bit of numerical exploration I found this to agree with $ - \ pi ^ 3/8 $ .

Answer 1

Primero configure $\sin t=u$ luego $u=\frac{2x}{1+x^2}$ y use el hecho de que $\sin^{-1}\left(\frac{2x}{1+x^2}\right)=\tan^{-1}(x)$ obtenemos

PS

donde la última integral se prueba aquí