# 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.

Let f and g be functions. Now consider:

- ∫
*a* f(*x*) + *b* g(*x*) *dx*

By the

sum rule in integration, this is:

- ∫ (
*a* f(*x*) *dx*) + ∫ (*b* g(*x*) *dx*)

By the

constant factor rule in integration, this reduces to:

*a* ∫ f(*x*) *dx* + *b* ∫ g(*x*) *dx*

Hence we have:

- ∫
*a* f(*x*) + *b* g(*x*) *dx* = *a* ∫ f(*x*) *dx* + *b* ∫ 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.