calculus - Calculate the upper sum $U_n$ and lower sum $L_n$, on a regular partition of the following interval.

As the title says, I am trying to calculate the general formulas for the following integral.
$$\int_0^1 (5+6x^2)dx$$

I've already completed a prior integral using the advice in this question, but I am having trouble following the same pattern here.

I have already calculated the following components, which are hopefully correct:

$$\Delta x = \frac 1n, \quad x_i = 1+i\Delta x, \quad f(x_i) = 5+6\left(1+\frac 1n\right)^2$$

Using this I was able to get to the following point: (1)

\begin{equation}
\begin{split}
\frac 1n \sum_{i=0}^n f(x_i)&=\frac 1n \sum_{i=0}^n 5+6(1+\frac 1n)^2 \\
&= \frac 1n (\sum_{i=0}^n 11 + \sum_{i=0}^n \frac {6i^2}{n^2} + \sum_{i=0}^n \frac {12i}{n})
\end{split}
\end{equation}

However my final answer after a few more steps was shown as incorrect. $$12n + \frac {6}{n^3}\frac n6 (2n+1)(n-1) + \frac{6(n+1)}{n}$$

I have obviously left out a few steps of my working, but I didn't want to bloat the question too much. Assuming my summations are correct at the end of (1), I then swapped things around to get the second and third as summations of $i$ and $i^2$, and then used the relevant rules to get my answer. I can edit to add all of my working if needed.

Answer

You have several mistakes in your first row of expressions. For the general expressions in my answer below, let the integral be $\int_a^b f(x)\,dx$.

Your $\Delta x$ is correct, being

$$\Delta x=\frac{b-a}n=\frac{1-0}n=\frac 1n$$

However, your $x_i$ is wrong and should be

$$x_i=a+i\Delta x=0+i\Delta x=i\Delta x$$

Your $f(x_i)$ is also wrong and should be

$$f(x_i)=5+6\left(\frac 1n\right)^2$$

Your formulas for $x_i$ and $f(x_i)$ seem to assume that the limits of integration are from $1$ to $2$, but the one you wrote is from $0$ to $1$.

Also, the limits on your summations are wrong. You have them from $0$ to $n$, but they should either be from $0$ to $n-1$ (as in the answer you linked to, which method is called the "left-side rule" or "left rectangular approximation method" or LRAM) or from $1$ to $n$ (which is the "right-side rule" or "right rectangular approximation method" or RRAM). So you want

$$\frac 1n\sum_{i=0}^{n-1} f(x_i) \qquad\text{or}\qquad \frac 1n\sum_{i=1}^{n} f(x_i)$$

Since your function is increasing, the lower sum $L_n$ will be the first summation and the upper sum $U_n$ will be the second.

Correct those errors and try again.