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If ∫f(x)dx exists, and if there is a function F(x) which has [(dF)/(dx)] (x) = f(x) in the range a ≤ x ≤ b, then

⌠ ⌡ b a  f(x)dx = F(b) − F(a)

ie, differentiation is the inverse of integration.

Real case: let F be continuously differentiable on [a,b] , with derivative F^′=f . Then ∫_a^b f exists, and

⌠ ⌡ b a  f = F(b)−F(a).

Complex case: let D be a domain and f : D → C be continuous. If f has an antiderivative F and ϕ: [a,b] → D is a path, then

⌠ ⌡ ϕ  f(z) dz = F( ϕ(b))−F( ϕ(a)).