Chapter 1.6, Problem 388E

The sine integral, defined as $S\left(x\right)={\displaystyle {\int }_0^x\frac{\sin t}{t}dt}$ is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large x. Show that for $k\ge 1,\left|S\left(2\pi k\right)-S\left(2\pi \left(k+1\right)\right)\right|\le \frac{1}{k\left(2k+1\right)\pi }$ . (Hint: $\sin \left(t+\pi \right)=-\sin t$ )