ABSTRACT
Elementary mathematics leaves the impression that there is marked difference between the two branches of calculus. Differentiation is a subject that is systematic: every evaluation is a consequence of a number of rules and some basic examples. However, integration is a mixture of art and science. The successful evaluation of an integral depends on the right approach, the right change of variables or a patient search in a table of integrals. In fact, the theory of indefinite integrals of elementary functions is complete [28, 29]. Risch’s algorithm determines whether a given function has an antiderivative within a given class of functions. However, the theory of definite integrals is far from complete and there is no general theory available. The level of complexity in the evaluation of a definite integral is hard to predict as can be seen in∫ ∞
e−x dx = 1, ∫ ∞ 0
dx =
√ π
2 , and
dx = Γ ( 4 3
) . (7.1.1)
The first integrand has an elementary primitive, the second one is the classical Gaussian integral, and the evaluation of the third requires Euler’s gamma
Volume
Γ(a) =
xa−1e−x dx. (7.1.2)
The table of integrals [40] contains a large variety of integrals. This paper continues the work initiated in [3, 63, 64, 65, 66, 67] with the objective of providing proofs and context of all the formulas in the table [40]. Some of them are truly elementary. In this paper we present a derivation of a small number of them.
