# Rules of Integration

# Linearity

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.

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