The limit of the integral \lim_{n \to \infty} \int_0^1 | \sin(nx)| \, dx evaluates to \frac{2}{\pi}. This conclusion is drawn from the periodic nature of |\sin x|, which occupies a consistent fraction of the area in any rectangle of the form [0, \pi]. As n increases, the area under |\sin(nx)| approaches this fraction in the unit square. The argument relies on partitioning the unit square into smaller rectangles that align with the sine function's periodicity. Ultimately, the area of the leftover segments diminishes to zero, confirming the limit.