Linearity of integration

In calculus, the integral of any linear combination of functions equals the same linear combination of the integrals of the functions; this property is known as linearity of integration.^[1] It is a fundamental property of the integral that encapsulates in a single rule two simpler rules of integration, the sum rule (the integral of the sum of two functions equals the sum of the integrals) and the constant factor rule (the integral of a constant multiple of a function equals a constant multiple of the integral).^[2] Linearity of integration is related to the linearity of summation, since integrals are thought of as infinite sums.

Statement and derivation

Let f and g be functions. Now consider:

[math]\displaystyle{ \int (af(x)+bg(x))\, dx. }[/math]

By the sum rule in integration, this is

[math]\displaystyle{ \int af(x)\, dx+\int bg(x)\, dx. }[/math]

By the constant factor rule in integration, this reduces to

[math]\displaystyle{ a\int f(x)\, dx+b\int g(x)\, dx. }[/math]

Hence we have

[math]\displaystyle{ \int (af(x)+bg(x))\, dx=a\int f(x)\, dx+b\int g(x)\, dx. }[/math]

It is also possible to infer linearity of integration as a consequence of linearity of differentiation.^[3]

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.^[4] 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.

References

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