Definite Integration

If f(x) is continuous on [a

If f(x) is continuous on [a,b] and φ(x) is its antiderivative, then ∫ₐᵇ f(x)dx = φ(b) - φ(a). The numbers 'a' and 'b' are called lower and upper limit

For a continuous function f(x) on

For a continuous function f(x) on [a,b]: ∫ₐᵇ f(x)dx = [F(x)]ₐᵇ = F(b) - F(a), where F'(x) = f(x). This connects differentiation and integration as inv

For symmetric intervals [

For symmetric intervals [-a,a]: If f(x) is even (f(-x) = f(x)), then ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx. If f(x) is odd (f(-x) = -f(x)), then ∫₋ₐᵃ f(x)dx = 0.

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b

∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx allows transformation of integrals by substituting x with (a+b-x). When combined with the original integral, this often le

∫ₐᵇ u dv = [uv]ₐᵇ

∫ₐᵇ u dv = [uv]ₐᵇ - ∫ₐᵇ v du. This extends the integration by parts method to definite integrals, useful for products involving logarithmic, inverse t