The authors discuss two approaches for obtaining approximate, validated formulas for the one-dimensional indefinite integral g ( x ) = ∫ax f ( t ) d t, with a ≤ x ≤ b. The formulas are validated in the sense that an interval-valued function G ( x ):=[ Ģ ( x ) , G ( x ) ] is given that satisfies Ģ ( x ) ≤ g ( x ) ≤ G ( x ) for every x ∈ [ a , b ].
The first approach is to find an inclusion of the integrand, that is an approximation for f ( t ) with a known error term, then integrate this inclusion to obtain an inclusion of the indefinite integral. The second approach finds an inclusion of the indefinite integral directly as a linear combination of function evaluations plus an interval-valued error term. This approach requires the use of a quadrature formula with a validated error term. The authors give an interesting example showing the application to the error function erf ( x ).
