Linearity Property of the Integral

The linearity property of the integral refers to the very important property of being able to break integrals apart into sums. That is, ∫(b,a) [f(x)dx + g(x)dx] = ∫ (b,a) f(x)dx + ∫(b,a)g(x)dx.

The content in today's blog is taken from Edwards and Penney's Calculus and Analytic Geometry.

Theorem: Linearity Property of Integrals

If α, β are constants and f(x) and g(x) are continuous functions on [a,b], then:

∫ (b,a) [αf(x) + βg(x)]dx = α∫(b,a)f(x)dx + β∫(b,a)g(x)dx

Proof:

(1) By the Constant Multiple Property (see Lemma 2, here):

∫(b,a) cf(x)dx = c∫(b,a) f(x)dx

(2) Let F(x) be the antiderivative of f(x) and G(x) be the antiderivative of g(x).

(3) d/dx[F(x) + G(x)] = f(x) + g(x) [See Lemma 3, here]

(4) Using the Fundamental Theorem of Calculus (see Thereom 2, here), this gives us:

∫ (b,a) [f(x) + g(x)]dx = [F(x) + G(x)] (b,a)

(5) Using the Evaluation of Integrals (see Theorem 3, here), we have:

[F(x) + G(x)](b,a) = [F(b) + G(b)] - [F(a) + G(a)] = [F(b) - F(a)] + [G(b) - G(a)] =

= [F(x)](b,a) + [G(x)](b,a) = ∫(b,a) f(x)dx + ∫(b,a) g(x)dx.

QED

References

• Edwards & Penney, Calculus and Analytic Geometry