Applications of Integration

A definite integral ∫ₐᵇ f(x)dx

A definite integral ∫ₐᵇ f(x)dx is defined as the limit of Riemann sums. The interval [a,b] is partitioned into n subintervals, and within each subinte

First Fundamental Theorem

First Fundamental Theorem: If F(x) = ∫ₐˣ f(u)du, then dF/dx = f(x). This shows that differentiation and integration are inverse operations. Second Fun

Key properties include

Key properties include: linearity (∫[αf + βg]dx = α∫f dx + β∫g dx), additivity over intervals (∫ₐᵇ + ∫ᵇᶜ = ∫ₐᶜ), symmetry properties for even and odd

Improper integrals have either infinite limits

Improper integrals have either infinite limits or discontinuous integrands. They are defined as limits: ∫ₐ∞ f(x)dx = lim_{t→∞} ∫ₐᵗ f(x)dx. An improper

Reduction formulae express integrals of higher

Reduction formulae express integrals of higher powers in terms of lower powers. For example, Iₙ = ∫₀^(π/2) sinⁿx dx = ((n-1)/n)Iₙ₋₂. By repeatedly app