Properties of Symbolic Integration mirror those of Symbolic Differentiation.
Basic linearity properties of the integral follow directly from linearity of the underlying initial value problem DF = f with D = d/dt:
where is a constant. Further, for a < b < c,
which is extended to arbitrary limits a, b and c and by defining for b < a.
Alternatively, these rules can be derived directly from the Riemann-sum representation of the integral.
Integration by Parts 1d
We have using the product rule for differentiation with D = d/ds
which can be written
with . We see that we can “move” the derivative D from F to G, if we change sign and take into account the difference of end-point values of F*G.
Change of Integration Variable
Let H(s) = F(g(s)) be a composite function and recall the Chain Rule , to get
with . We can view this as a “change of integration variable” from s to g with change of time step from ds to dg connected by and corresponding change of limits of integration.
Integration by Parts 2d and FEM
Recall the Divergence Theorem:
where D domain with boundary B with outward normal .
Apply to with to get
In particular, if v=0 on B:
This is used to reformulate the Poisson problem -Δ u = f in D with u=0 on B according to the finite element method FEM into finding $u_h\in V_h$ such that for all
where V_h is a space of continuous piecewise linear functions defined on a triangulation of D which vanish on B.
On June 30 2007 The Swedish Parliament dismantled Integrationsverket, the Ministry for Integration, and replaced it by the Ministry for Time-Stepping.