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Integrals · Integral Properties

Linearity of Integrals

∫ab[αf(x)+βg(x)] dx=α∫abf(x) dx+β∫abg(x) dx\int_a^b [\alpha f(x) + \beta g(x)]\, dx = \alpha \int_a^b f(x)\, dx + \beta \int_a^b g(x)\, dx

Integrals are linear: constants factor out and the integral of a sum is the sum of integrals.

  • 2 worked examples
  • 19 in Integrals

Worked examples

Evaluate ∫₀¹ (3x² + 2x) dx.
  1. 3∫₀¹ x² dx + 2∫₀¹ x dx = 3[x³/3]₀¹ + 2[x²/2]₀¹ = 3(1/3) + 2(1/2) = 1 + 1

Answer: 2

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Evaluate ∫₀^π (2 sin x - 3 cos x) dx.

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Common questions

What is the Linearity of Integrals?

Integrals are linear: constants factor out and the integral of a sum is the sum of integrals. It is one of the integrals formulas in the CalcRef reference.

How do you use the Linearity of Integrals?

Worked example. Evaluate ∫₀¹ (3x² + 2x) dx. 3∫₀¹ x² dx + 2∫₀¹ x dx = 3[x³/3]₀¹ + 2[x²/2]₀¹ = 3(1/3) + 2(1/2) = 1 + 1. Answer: 2.

Practice

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