Definite Integration — Formula Sheet

If f(x) is continuous on [a,b] and F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x)dx = F(b) - F(a)

Notation

∫ₐᵇ f(x)dx = [F(x)]ₐᵇ where 'a' is lower limit and 'b' is upper limit

Apply integration by parts

∫udv = [uv]ₐᵇ - ∫vdu evaluated from a to b

Property 1

∫ₐᵃ f(x)dx = 0 (integral from a point to itself is zero)

Property 2

∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx (reversing limits changes sign)

Property 3

∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx (additivity property)

If f(x) is continuous on [a,b] and φ(x) is its antiderivative, then ∫ₐᵇ f(x)dx = φ(b) - φ(a).

For a continuous function f(x) on [a,b]

∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) - F(a), where F'(x) = f(x).

For symmetric intervals [-a,a]

If f(x) is even (f(-x) = f(x)), then ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx.

If f(x) is odd (f(-x) = -f(x)), then ∫₋ₐᵃ f(x)dx = 0.

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx allows transformation of integrals by substituting x with (a+b-x).

∫ₐᵇ u dv = [uv]ₐᵇ - ∫ₐᵇ v du.