Basic linearity properties of the integral follow directly from linearity of the underlying IVP :
where is a constant. Further, for ,
which is extended to arbitrary limits and , by defining for
Alternatively, these rules can be derived directly from the Riemann-sum representation of the integral.
Integration by Parts
By the Fundamental Theorem of Calculus, we have using the product rule for differentiation:
which can be written
with . We see that we “can move the dot” from $u$ to $v$ if we change sign and take into account the difference of end-point values of .
Change of Integration Variable
If is differentiable and is Lipschitz continuous, then
because with , we have .
On June 30 2007 The Swedish Parliament dismantled Integrationsverket, the Ministry for Integration, and replaced it by the Ministry for Time-Stepping.