Definite Integral Properties: Rules, Proofs, Examples

Sometimes a definite integral is hard to compute directly but easy after a clever rewrite. The properties of definite integrals are a toolkit of such rewrites: splitting intervals, reversing limits, shifting variables, and exploiting symmetry. Each property is a small theorem with a one-line proof from the limit-of-sums definition, and together they solve many integrals that resist brute force.

These properties matter enormously in exams because the hardest definite-integral questions are designed around them. An integral like ∫₀^{π/2} sin x/(sin x + cos x) dx looks impossible until the property ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx halves the work.

Key Concepts

• Interval splitting: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫_c^b f(x) dx for any c between a and b.
• Reversal: ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx; in particular ∫ₐᵃ f(x) dx = 0.
• Linearity: ∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f ± ∫ₐᵇ g; constants factor out.
• King property: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx — replace x by (a + b − x).
• Special cases: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx; ∫₀^{2a} f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a − x) dx.
• Doubling/halving: ∫₀^{2a} f(x) dx = 2∫₀ᵃ f(x) dx if f(2a − x) = f(x), and 0 if f(2a − x) = −f(x).
• Symmetric limits: ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even, 0 if f is odd.

Formulas

• ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫_c^b f(x) dx
• ∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx
• ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx
• ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
• ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx (f even); = 0 (f odd)

Worked Example

Evaluate ∫₀^{π/2} sin x/(sin x + cos x) dx — call it I. Step 1: apply the King property with a + b = π/2: I = ∫₀^{π/2} sin(π/2 − x)/[sin(π/2 − x) + cos(π/2 − x)] dx = ∫₀^{π/2} cos x/(cos x + sin x) dx. Step 2: add the two forms: 2I = ∫₀^{π/2} (sin x + cos x)/(sin x + cos x) dx = ∫₀^{π/2} 1 dx = π/2. Step 3: divide: I = π/4. A problem no antiderivative could touch collapses in three lines.

Common Misconceptions

• Applying ∫₋ₐᵃ = 0 to non-odd functions: ∫₋₁¹ x² dx is 2/3, not 0 — the zero shortcut needs f(−x) = −f(x) exactly.
• Reversing limits without the minus sign: ∫ₐᵇ = −∫ᵇₐ; forgetting the sign flips the answer.
• Using the King property with wrong bounds: the replacement is (a + b − x) with the actual limits a and b, not always (π/2 − x) or (a − x).

Real-World Applications

Symmetry properties halve the work in computing centres of mass, moments of inertia and electric fields of symmetric charge distributions — physicists invoke even/odd arguments constantly. Signal processing uses the King property to relate time-reversed signals. Numerical integration exploits symmetry to cut computation in half.

Definite-integral properties are strategic tools: split, reverse, shift, mirror. When an integral looks impossible, one of these properties is usually the intended door — and the even/odd and King properties open the most doors of all.