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Integrals · Integral Properties

Additivity of Integrals

∫abf(x) dx+∫bcf(x) dx=∫acf(x) dx\int_a^b f(x)\, dx + \int_b^c f(x)\, dx = \int_a^c f(x)\, dx

The integral over [a,c] can be split at any point b into two integrals.

  • 2 worked examples
  • 19 in Integrals

Worked examples

If ∫₀² f(x) dx = 3 and ∫₂⁵ f(x) dx = 7, find ∫₀⁵ f(x) dx.
  1. ∫₀⁵ f(x) dx = ∫₀² f(x) dx + ∫₂⁵ f(x) dx = 3 + 7

Answer: 10

1 more worked examplePremium

If ∫₀³ f(x) dx = 8 and ∫₀⁷ f(x) dx = 5, find ∫₃⁷ f(x) dx.

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Common questions

What is the Additivity of Integrals?

The integral over [a,c] can be split at any point b into two integrals. It is one of the integrals formulas in the CalcRef reference.

How do you use the Additivity of Integrals?

Worked example. If ∫₀² f(x) dx = 3 and ∫₂⁵ f(x) dx = 7, find ∫₀⁵ f(x) dx. ∫₀⁵ f(x) dx = ∫₀² f(x) dx + ∫₂⁵ f(x) dx = 3 + 7. Answer: 10.

Practice

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