Express Integral as a Riemann Sum and solve

Hello MHF,

I have been able to process this question a little bit, but got a bit stuck when i got to a certain point..

Express the integral as a limit of sums;

$\displaystyle \int_{0}^{\pi}\sin5x \ dx$

So by setting $\displaystyle \Delta x=\frac{\pi}{n}$ and $\displaystyle x_{i}=\frac{\pi i}{n}$

Then, $\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\sin(\frac{5\pi i}{n})*\frac{\pi}{n}$

$\displaystyle \lim_{n\to\infty}\frac{\pi}{n}\sum_{i=1}^{n}\sin(\ frac{5\pi i}{n})

$

$\displaystyle \lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{n}\ sum_{i=1}^{n}i)

$

$\displaystyle \lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{n}* \frac{n^2+n}{2})

$

$\displaystyle \lim_{n\to\infty}\frac{\pi}{n}\sin(\frac{5\pi}{2}* (1+n))

$

But form here I dont know if i have done something incorrectly, or how to keep reducing it to get rid of n..