When I was introduced to integration, I was briefly exposed to the definition of a Riemann sum, and as an illustration we worked out a couple of definite integrals directly from this definition: $$ \int\limits_a^b f\left(x\right)\,dx = \lim_{n\rightarrow\infty} \sum\limits_{k=0}^{n} f\left(x_k\right) \Delta x, \qquad \Delta x = \frac{b-a}{n}, \qquad x_k = a + k\Delta x $$ These turned out to be fairly tedious, and I was quickly shown that the Fundamental Theorem of Calculus (FTC) provides a much more convenient way of doing things.
I am wondering if there are any examples of definite integrals that are hard to evaluate using the FTC because the antiderivative of $f(x)$ is hard to find, yet are amenable to being worked out directly from the definition.
Let $C$ be the Cantor set. Define $$f(x)=\begin{cases}1 & x\not \in C\\ 0&x \in C\\ \end{cases}$$
Then $\int_0^1 f(x)dx =1$.
(And is well defined as a Riemann integral.)