Although we implicitly use the definition of the Riemann Sums every time we cacluate a definite integral is it actually possible to calculate the majority of integrals via Riemann Sums?

Example:

$\displaystyle \int_0^1 x~dx$ this is easily able to be calculated using the rieman formula $\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}f\left(M_i\right)\d elta x_i$ (I omit the details)

But what about another fairly elementary integral? How would we go about using the above defintion to find say

$\displaystyle \int_1^e \ln(x)~dx$?

Is it actually feasible using just summing techniques (besides obviously integrating) to calculate $\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}\ln\left(1+\frac{e-1}{n}i\right)\cdot\frac{e-1}{n}$?

Even if someone could do the above sum (which Im sure someone can) what about a harder one like $\displaystyle \int_0^1 e^x\sin(x)~dx$. And what about all of this using the formal Riemann-Stieltjes definitions?

A similar argument brings us to ask if we can practically find say

$\displaystyle \lim_{x\to 0}\frac{\tan(x)-\sin(x)}{x^3}$ using only $\displaystyle \delta-\varepsilon$ defintions?

Note: I am not asking whether or not we should use these defintions to find integrals and limits. What I am asking is if someone said compute $\displaystyle \int_0^1 e^x\sin(x)~dx$ using only Riemann sums would it be feasible?

Any input/discussion would be appreciated :)