Some authors maintain a distinction between antiderivatives and indefinite integrals.

Assuming the curve is smooth within a neighborhood, this generalizes to indefinite integrals:

Indefinite integrals are called indefinite because their solutions are only unique up to a constant.

The feature that the numeric lacks is the ability to solve algebraic equations such as indefinite integrals and derivatives.

Tables of Indefinite Integrals.

In these cases, it is not possible to evaluate indefinite integrals, but definite integrals can be evaluated numerically, for instance by Simpson's rule.

Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions, typically using a computer algebra system.

There are yet other cases where definite integrals can be evaluated exactly without numerical methods, but indefinite integrals cannot, for lack of an elementary antiderivative.

As with many properties of integrals in calculus, the sum rule applies both to definite integrals and indefinite integrals.

Passing from the case of indefinite integrals to the case of integrals over an interval [a,b], we get exactly the same form of rule (the arbitrary constant of integration disappears).

There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.

However, the Risch algorithm applies only to indefinite integrals and most of the integrals of interest to physicists, theoretical chemists and engineers, are definite integrals often related to Laplace transforms, Fourier transforms and Mellin transforms.