Time Series

∫ₐᵃ f(x) dx = 0, ∫ᵇₐ f(x) dx = -∫ₐᵇ f(x) dx, ∫ᵇₐ f(x) dx = ∫ᶜₐ f(x) dx + ∫ᵇᶜ f(x) dx, ∫ᵇₐ f(x) dx = ∫ᵇₐ f(t) dt

All four are fundamental properties: (1) Same limits give zero, (2) Reversing limits changes sign, (3) Splitting interval property, (4) Variable independence property

f(-x) = f(x), ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx, Graph is symmetric about y-axis

For even functions: f(-x) = f(x), symmetric about y-axis, and integral over symmetric interval equals twice the integral from 0 to a. The property f(-x) = -f(x) applies to odd functions.

2

Step 1: Rewrite 1/√x as x^(-1/2) Step 2: Antiderivative is x^(1/2)/(1/2) = 2√x Step 3: Apply limits: [2√x]₁⁴ = 2√4 - 2√1 = 2(2) - 2(1) = 4 - 2 = 2

8

Using the additive property of definite integrals: ∫₀² f(x) dx = ∫₀¹ f(x) dx + ∫₁² f(x) dx = 3 + 5 = 8