The basic mechanism for evaluating a definite integral is the Fundamental Theorem of Calculus (FTC), formally stated as Theorem 4.3.1.

The first conclusion of the FTC is that a definite integral, the limit of a Riemann sum, can be evaluated by subtracting the endpoint values of any antiderivative. That is the meaning of the symbols in the first conclusion of the theorem, where $F$ is any antiderivative of the integrand $f$, and the value of the definite integral is $F\left(b\right)-F\left(a\right)$.

The second conclusion of the FTC is that a definite integral with a varying endpoint defines a function whose derivative is the integrand. In other words, the integral in hypothesis (3) defines an antiderivative, so that  integration (the limit of a Riemann sum) and differentiation are inverse operations.

Because of the FTC, and only because of the FTC, the integral sign is used to denote the operation of finding an antiderivative. Thus, the antiderivative is identified with the indefinite integral, denoted by $\int f \mathit{\ⅆ}x$.

For the functions $f_k$ in Table 4.3.1, Maple computes their definite integrals on $\left[a\,b\right]$ by taking the limit of a right Riemann sum. The results are consistent with finding an antiderivative $F_k$ and evaluating $F_k\left(b\right)-F_k\left(a\right)$, that is, by using the FTC.

The first four functions all yield to the "Power rule for antidifferentiation," that is, to the rule "add 1 to the power and divide by the new power." (In symbols, this is the third entry in Table 3.10.1.) Antiderivatives for the exponential and trig functions can also be found in Table 3.10.1.

If $F$ is an antiderivative of $f$, a common notation for the difference $F\left(b\right)-F\left(a\right)$ that arises in the evaluation of the definite integral ${\int }_a^{b}f\left(x\right) \mathit{\ⅆ}x$ is $F {\|}_a^b$. (Sometimes, the stroke is replaced by "$\]$", the closing square bracket.) Although such notation can sometimes be written in Maple, it is not part of the code structure used when a definite integral is evaluated by Maple.