Using the upper and lower sums, I've tried to solve $\int_1^e\frac{1}{x}dx$ in the following way.
Let $$\Delta x = \frac{\mathit e -1}{n}$$ and let a partition $P$ be given by $$ P = \{1,1+\Delta x, 1+2\Delta x,...,1+(n-1)\Delta x,\mathit e\}$$
Since $1/x$ is a monotonically decreasing function, noting the definition of the upper and lower partitions is $M_i=\mathrm {sup}\{f(x) \lvert \text{P is a partition} \}$ and $m_i=\mathrm {inf}\{f(x)\lvert \text{P is a partition} \}$, this means $$M_i = f(x_{i-1}) = \frac{a}{1+\frac{(i-1)(e-1)}{n}} \quad \mathrm {and} \quad m_i = f(x_i) = \frac{a}{1+\frac{i(e-1)}{n}}$$
So the upper sum is $$U(f,P) = \sum_{i=1}^n M_i\Delta x = \sum_{i=1}^n \frac{e-1}{n+(i-1)(e-1)}$$ and the lower sum is $$L(f,P)=\sum_{i=1}^n m_i\Delta x=\sum_{i=1}^n\frac{e-1}{n+i(e-1)}$$
But I can't evaluate these sums. I'm not sure if my process is wrong, so I am getting difficult sums or what.