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Newton–Leibniz formula

For continuous f and an antiderivative F, the integral of f over [a,b] equals F(b)-F(a).

Let f:[a,b]→Rf:[a,b]\to\mathbb{R} be . If F:[a,b]→RF:[a,b]\to\mathbb{R} is continuous on [a,b][a,b], differentiable on (a,b)(a,b), and satisfies F′(x)=f(x)F'(x)=f(x) for every x∈(a,b)x\in(a,b), then

∫abf(x) dx=F(b)−F(a).\int_a^b f(x)\,dx = F(b)-F(a).
Remarks

This is the evaluation form of the .