Can we use (higher order) derivatives to help integration? I'm wondering if it's possible to use derivatives to ease the evaluation of an integral.  For instance, I know that to evaluate an integral with enough precision I need to evaluate it at $n$ points.  I also have a method to get higher-order derivatives, like the 2nd derivative, 3rd derivative, etc. with relative ease.  So I'm wondering, and hopefully this isn't to vague, if I can make use of these derivatives so that I can evaluate the function at less than $n$ points.
In other words, can I use derivatives so that I can combine them with an integral somehow so that I can evaluate the integral at fewer points?  I know this may be vague, but I'm just trying to get a general feel of whether derivatives can with the evaluation of an integral.
 A: Sure. You could, for instance, use a Taylor series expansion of the function, then integrate the resulting polynomial very easily.

Say you want to evaluate $\int_a^b f(x)\ dx$, and $f \in C^k(\Bbb R)$ for some $k$ sufficiently large.
Then, for some $c \in [a,b]$, the Taylor expansion of $f$ is:
$$f(x) = f(c) + f'(c)(x-c) + \frac{f'(c)}{2!}(x-c)^2 + \cdots + \frac{f^{(k)}(c)}{k!}(x-c)^k + o(x^{k+1}).$$
Suppose the error term is sufficiently small. Then,
$$\begin{align*}
\int_a^b f(x)\ dx &= \epsilon +  \int_a^b f(c) + f'(c)(x-c) + \frac{f'(c)}{2!}(x-c)^2 + \cdots + \frac{f^{(k)}(c)}{k!}(x-c)^k\ dx \\
 &= \epsilon + f(c)\int_a^b dx + f'(c)\int_a^b (x-c)\ dx + \frac{f''(c)}{2!} \int_a^b (x-c)^2\ dx + \cdots
\end{align*}$$
Polynomials are easy to integrate; you can do so exactly. Then, you have your integral.
In practice, this is rarely done, because evaluating the derivative is typically expensive, inexact, or both, or the Taylor series does not converge nicely, or rapidly enough.
However, this is the basis for deriving a great many numerical techniques. There are certainly better ways to control the error.
