Answer
$$ 0.347 $$
Work Step by Step
Given $$ \int_{0}^{\pi / 4} \tan \theta d \theta, \quad N=10 $$
Since $\Delta x=\dfrac{b-a}{N}=\dfrac{\pi}{40}$ , then by using Simpson’s rule
\begin{align*}
S_{n}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3).....+4f(x_{n-1})+f(x_n)\right]\\
S_{10}&=\dfrac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5) +2f(x_6)+4f(x_7) +2f(x_8)+4f(x_9) +f(x_{10}) \right] \\
&=\dfrac{\pi}{120}\left[f(0)+4f(\pi/40)+2f(2\pi/40)+4f(3\pi/40)+2f(4\pi/40) +4f(5\pi/40)+2f(6\pi/40)+4f(7\pi/40)+2f(8\pi/40) +4f(9\pi/40)+f(\pi/4) \right]\\
&\approx 0.347
\end{align*}
