Time Series — Formula Sheet

Definite integral ∫ᵇₐ f(x) dx represents the area under curve y = f(x) between x = a and x = b

Formula

∫ᵇₐ f(x) dx = lim(n→∞) Σᵣ₌₁ⁿ h·f(a + rh) where h = (b-a)/n

If F(x) is antiderivative of f(x), then ∫ᵇₐ f(x) dx = F(b) - F(a)

Property I

∫ₐₐ f(x) dx = 0 (integral over zero width interval)

Property II

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

Property III

∫ᵇₐ f(x) dx = ∫ᵇₐ f(t) dt (integration variable doesn't matter)

Property VI

∫₀ₐ f(x) dx = ∫₀ₐ f(a - x) dx (special case for [0,a])

The definite integral ∫ᵇₐ f(x)dx is defined as the limit of Riemann sums.

If F(x) is an antiderivative of f(x), then ∫ᵇₐ f(x)dx = F(b) - F(a) = [F(x)]ᵇₐ.

For even functions f(-x) = f(x), ∫₋ₐₐ f(x)dx = 2∫₀ₐ f(x)dx.

For odd functions f(-x) = -f(x), ∫₋ₐₐ f(x)dx = 0.

For integrals of the form ∫₀^π/2 sinⁿx dx and ∫₀^π/2 cosⁿx dx: If n is odd: result = (n-1)/n × (n-3)/(n-2) × (n-5)/(n-4) × ... × 4/5 × 2/3.