Applications of Integration — Formula Sheet

Three evaluation rules for equal-width partitions

Left-end rule (ξ_i = x_{i-1}), Right-end rule (ξ_i = x_i), Mid-point rule (ξ_i = midpoint).

First Fundamental Theorem

If F(x) = ∫[a to x] f(u)du, then d/dx[F(x)] = f(x). Integration and differentiation are inverse operations.

Second Fundamental Theorem

∫[a to b] f(x)dx = F(b) - F(a), where F(x) is any anti-derivative of f(x). Written as [F(x)]_a^b.

Notation

u⁽¹⁾ = du/dx, u⁽²⁾ = d²u/dx², ... (successive derivatives); v₍₁₎ = ∫v dx, v₍₂₎ = ∫v₍₁₎dx, ... (successive integrals).

Definition

∫[a to ∞] f(x)dx = lim(t→∞) ∫[a to t] f(x)dx.

Definition

∫[-∞ to a] f(x)dx = lim(t→-∞) ∫[t to a] f(x)dx.

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

First Fundamental Theorem

If F(x) = ∫ₐˣ f(u)du, then dF/dx = f(x).

They are defined as limits

∫ₐ∞ f(x)dx = lim_{t→∞} ∫ₐᵗ f(x)dx.

For example, Iₙ = ∫₀^(π/2) sinⁿx dx = ((n-1)/n)Iₙ₋₂.