Chapter 5 Conceptual Project:

The topic of this project is the so-called sine integral function, which is important for its applications, most notably in electrical engineering and signal processing.

1. Consider the following piecewise‑defined function.

   $f\left(t\right)=\{\begin{array}{ll}{\displaystyle \frac{\sin t}{t}}& \text{if }t>0\hfill \\ 1& \text{if }t=0\hfill \end{array}$

   Prove that for any $x\ge 0$, $f\left(t\right)$ is integrable on $\left[0,x\right]$.

2. The sine integral function is defined as follows.

   $\mathrm{Si}\left(x\right)={\displaystyle {\int }_0^{x}f\left(t\right)\mathit{d}t}$, for $x\ge 0$

   Prove that $\mathrm{Si}\left(x\right)$ is continuous.

3. Find the derivative ${\displaystyle \frac{\mathit{d}}{\mathit{d}x}}\mathrm{Si}\left(x\right)$.

4. Without graphing first, write a short paragraph on why you would expect the graph of $\mathrm{Si}\left(x\right)$ to be oscillating. Explain why its amplitude is expected to decrease as $x\to \infty$.

5. Find the x‑values where the relative maxima and minima of $\mathrm{Si}\left(x\right)$ occur.

6. Extend the definition of $\mathrm{Si}\left(x\right)$ to negative x‑values and prove that for any $a>0$, ${\displaystyle {\int }_{-a}^a\mathrm{Si}\left(x\right)\mathit{d}x}=0$.

7. Use a graphing utility to plot the graph of $\mathrm{Si}\left(x\right)$ on the interval $\left[-8\pi ,8\pi \right]$.

8. Use a graphing utility to approximate the range of $y=\mathrm{Si}\left(x\right)$ to four decimal places.