SE,
I've been mulling over the following problem, which involves using Riemann sums to evaluate
$\int_{0}^1 (x^4+\cos(x)) dx$
And I am a little stuck. Here is my attempt at a solution: I've put the step numbers in for reference
I would really appreciate some guidance - I feel like I am at the cusp of a solution.
Your work is correct through most of the steps.
The following should help you finish.
$$\begin{align}\frac{1}{n}\sum_{i=1}^n\cos\left(\frac{i}{n}\right) &= \frac{\sin\left(\frac{n}{2}\frac{1}{n}\right)\cos\left(\frac{n+1}{2}\frac{1}{n}\right)}{n\sin\left(\frac{1}{2}\frac{1}{n}\right)} \\ &= \frac{2\sin\left(\frac{1}{2}\right)\cos\left(\frac{1}{2}+\frac{1}{2n}\right)}{2n\sin\left(\frac{1}{2n}\right)} \end{align}$$
Since,
$$\lim_{n \to \infty} 2n\sin\left(\frac{1}{2n}\right)= \lim_{n \to \infty} \frac{\sin\left(\frac{1}{2n}\right)}{\frac{1}{2n}} = 1,$$
you can conclude
$$\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^n\cos\left(\frac{i}{n}\right) = 2 \sin(1/2)\cos(1/2) = \sin(1).$$