Definite Integrals

The definite integral ∫ₐᵇ f(x)dx

The definite integral ∫ₐᵇ f(x)dx is defined as the limit of the sum of areas of rectangles under the curve as the width of each rectangle approaches z

If f(x) is continuous on [a

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). This theorem connects differentiation and integra

Key properties include

Key properties include: ∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx, ∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx, and symmetry properties like ∫₋ₐᵃ f(x)dx = 0 if f is odd, and

The definite integral ∫ₐᵇ f(x)dx gives

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

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. For even n, t