Indefinite and definite integrals. Methods of integration

Indefinite integral $\displaystyle\int f(x)\,dx$

The symbol $\displaystyle\int f(x)\,dx$ denotes the set of all antiderivatives of $f$ on the interval under consideration. One also writes $\displaystyle\int f(x)\,dx = F(x)+C,$ if $F'=f$ wherever the integral is considered. Definite integral: if $f$ is continuous on $[a;b]$ and $F$ is any antiderivative on $(a;b)$ extended continuously to $[a;b]$ , then $\displaystyle\int_a^b f(x)\,dx=F(b)-F(a).$ The right-hand side is often written $\left.F(x)\right|_a^b$ .

Newton–Leibniz links accumulating $f$ (“area” with sign) and the difference $F(b)-F(a)$

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