$\displaystyle \text{Riemannian sum, limit, and integration:}$

$\displaystyle \int_{\pi}^{2\pi}\cos(x)dx$

$\displaystyle \begin{align*}\displaystyle \Delta x &= \frac{b -a}{n}\\ &= \frac{2\pi - \pi}{n}\\

&= \frac{\pi}{n}

\end{align*}$

$\displaystyle \begin{align*}x_{i} &= a + \Delta x . i\\

&= \pi + \frac{\pi i}{n}

\end{align*}$

$\displaystyle \begin{align*}

&\lim_{n \to \infty} \sum_{i = 1}^{n} \Delta x . \cos(x_{i})dx\\

&=\lim_{n \to \infty} \sum_{i = 1}^{n} \frac{\pi}{n} . \cos(\pi + \frac{\pi i}{n})dx

\end{align*}$

How can I evaluate the above expression and get 0? I know how to get this value using integration. But I want to know practically how to arrive at that value?