Calculate, with accuracy 0.01: integral from 0 to ² (sin
Calculate, with accuracy $0.01$: $\displaystyle\int_0^2\frac{\sin x}{x}\,dx$.
Explanation of Method
This is the sine integral $\mathrm{Si}(2)$, which has no elementary closed form. As with the Catalan's-constant integral, the integrand has a removable singularity at $x=0$ (limit $1$), so setting $f(0)=1$ gives a smooth function that Simpson's rule handles efficiently with very few subintervals.
Step 1: Handling the singularity at $x=0$
From $\sin x=x-\dfrac{x^3}{3!}+\cdots$, $\dfrac{\sin x}{x}\to1$ as $x\to0$, so set $f(0)=1$.
Step 2: Choose a subinterval count
A conservative bound on $|f^{(4)}|$ over $[0,2]$ (order $10$) combined with the Simpson error formula shows $2n=4$ subintervals are already more than enough for the required $0.01$ accuracy.
Step 3: Apply Simpson's formula, $h=0.5$
| $x_i$ | 0 | 0.5 | 1.0 | 1.5 | 2.0 |
| $f(x_i)$ | 1.000000 | 0.958851 | 0.841471 | 0.664997 | 0.454649 |
| weight | 1 | 4 | 2 | 4 | 1 |
Weighted sum $S\approx9.632982$:
\[I_{\text{Simpson}}=\frac{0.5}{3}\times9.632982\approx1.605497.\]
\[\boxed{I_{\text{Simpson}}\approx1.605}\ \text{(to accuracy }0.01\text{)}\]
Step 4: Accuracy check
The known value is $\mathrm{Si}(2)\approx1.605413$, so the actual error is $\approx8.4\times10^{-5}$, well within the required $0.01$ — and a repeat with $2n=6$ intervals gives essentially the same result, confirming convergence.
Step 5: Remark
The sine integral $\mathrm{Si}(x)$ appears throughout signal processing, diffraction, and antenna theory; the sinc-type integrand $\sin x/x$ is smooth on $[0,2]$ once the removable singularity at the origin is handled, making it an ideal candidate for Simpson's rule.
Result
Final answer:
\[\boxed{\int_0^2\frac{\sin x}{x}\,dx\approx1.605\ (\text{true }\mathrm{Si}(2)\approx1.60541),\ \ E\approx8\times10^{-5}}\]