Properties of the integral and linearity

Linearity of the integral

For any continuous functions f and g on [a;b], and any real number k:

∫_a^b (f+g)(x) dx = ∫_a^b f(x) dx + ∫_a^b g(x) dx

∫_a^b k*f(x) dx = k * ∫_a^b f(x) dx

Example: ∫₀¹ (3x² + 2) dx = 3 × ∫₀¹ x² dx + 2 × ∫₀¹ 1 dx = 3 × (1/3) + 2 × 1 = 1 + 2 = 3.

Chasles’s relation

For any real number c:

∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx

This relation allows us to break down the calculation of an integral, particularly when f changes sign over the interval under consideration.

Integrals and Order

If f ≤ g on [a;b], then ∫_a^b f(x) dx ≤ ∫_a^b g(x) dx. In particular, if f ≥ 0 on [a;b], then the integral from a to b of f(x) dx ≥ 0.

Common pitfall

Chasles’s relation remains valid even if c is not between a and b, but be careful with the order of the limits: ∫_a^b f(x) dx = -∫_b^a f(x) dx. Reversing the limits changes the sign of the result.