Evaluate $\int_0^\pi \frac {\sin x}{x} \,dx$ using Simpson's $\frac {1}{3}$ rule and $\frac {3}{8}$ rule with $n=6$

Question

Evaluate $\displaystyle\int_0^\pi \frac {\sin x}{x}\,dx$ using Simpson's $\dfrac {1}{3}$ rule and $\dfrac {3}{8}$ rule with $n=6$.

For $n=6$, $h=\dfrac {\pi - 0}{6}=\dfrac {\pi}{6}$.

But, the value of $f(x)=\dfrac {\sin x}{x}$ at $x=0,\pi$ takes $\dfrac {0}{0}$ form. How to solve this ?

Answer 1

Just consider the function $$ f(x)=\begin{cases}\dfrac{\sin x }{x},& x \ne 0\\ 1, & x=0 \end{cases}, $$

which is continuous in $\mathbb{R}$.