Linearity of integration

In calculus, linearity is a fundamental property of the integral that follows from the sum rule in integration and the constant factor rule in integration. Linearity of integration is related to the linearity of summation, since integrals are thought of as infinite sums.

Let f and g be functions. Now consider:

:$int\; (af(x)+bg(x)),\; dx.$

By the sum rule in integration, this is

:$int\; af(x),\; dx+int\; bg(x),\; dx.$

By the constant factor rule in integration, this reduces to

:$aint\; f(x),\; dx+bint\; g(x),\; dx.$

Hence we have

:$int\; (af(x)+bg(x)),\; dx=aint\; f(x),\; dx+bint\; g(x),\; dx.$

Operator notation

The differential operator is linear — if we use the Heaviside D notation to denote this, we may extend D⁻¹ to mean the first integral. To say that D⁻¹ is "therefore" linear requires a moment to discuss the arbitrary constant of integration; D⁻¹ would be straightforward to show linear if the arbitrary constant of integration could be set to zero.

Abstractly, we can say that D is a linear transformation from some vector space "V" to another one, "W". We know that D("c") = 0 for any constant function "c". We can by general theory (mean value theorem)identify the subspace "C" of "V", consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D⁻¹ is a well-defined linear transformation that is bijective on Im D and takes values in "V"/"C".

That is, we treat the "arbitrary constant of integration" as a notation for a coset "f" + "C"; and all is well with the argument.