Definite Integrals

∫ₐᵇ f(x) dx = F(b) − F(a)

∫ₐᵇ f(x) dx = −∫ᵇₐ f(x) dx

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

∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx

∫₀²ᵃ f(x) dx = ∫₀ᵃ f(x) dx + ∫₀ᵃ f(2a−x) dx

∫₀^{π/2} sinⁿx dx = (n−1)/n × (n−3)/(n−2) × ⋯ × π/2 (n even)

∫₀^{π/2} sinⁿx dx = (n−1)/n × (n−3)/(n−2) × ⋯ × 2/3 (n odd)

∫₀^{π/2} sinᵐx cosⁿx dx = [(n−1)/(m+n)] × [(n−3)/(m+n−2)] × ⋯

Area = ∫ₐᵇ f(x) dx

Area = ∫ₐᵇ |f(x)| dx

Area between curves = ∫ₐᵇ [f(x) − g(x)] dx