Prove one of the following....

[DreamWeaver]

There's a unified pattern at work here...
Problem 1:

$$\int_0^{\infty}\frac{\log(1-\sin x)}{x}\,dx=-\frac{\pi^2}{6}=-\zeta(2)$$
Problem 2:

$$\int_0^{\infty}\frac{ \text{Li}_m(\sin x) }{x}\,dx=\text{Li}_{m+1}(1)=\zeta (m+1)$$

 

[Aaron Bex]

The unified pattern is that both problems are related to the evaluation of integrals involving the logarithm and the polylogarithm. The integrals can be evaluated in terms of the Riemann zeta function, with the result being a simple expression in terms of the zeta function for each problem.