Definite Integrals — Formula Sheet

∫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

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

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

Area = ∫ₐᵇ f(x) dx

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

The definite integral ∫ₐᵇ f(x) dx is defined as the limit of a sum of areas of rectangles under the curve.

Divide the interval [a, b] into n equal parts, each of width h = (b−a)/n.

The width of each subinterval is h = (b - a)/n.

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

The definite integral ∫ₐᵇ f(x)dx gives the area bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b, provided f(x) ≥ 0 in [a,b].

For integrals like ∫₀^{π/2} sinⁿx dx or ∫₀^{π/2} sinᵐx cosⁿx dx, reduction formulas provide a pattern based on whether n is even or odd.