Teorema fondamentale del calcolo integrale

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$$\begin{gathered} F_0= \\ ={\int }_o^{x}f(t)\cdot dt\to F^{{}^{\prime }}(x)= \\ =\underset{h\to 0}{\lim }\frac{F(x+h)-F(x)}{h}= \\ =\underset{h\to 0}{\lim }\frac{1}{h}\cdot \left\{\right({\int }_o^{x+h}fdt)-({\int }_o^{x}fdt\left)\right\}= \\ =\frac{1}{h}\left\{\right({\int }_o^{x}fdt)+({\int }_x^{x+h}fdt)-({\int }_o^{x}fdt\left)\right\}= \\ =\underset{h\to 0}{\lim }\frac{{\int }_x^{x+h}f(t)\cdot dt}{h}= \\ =\underset{h\to 0}{\lim }\frac{T}{h}= \\ =\underset{h\to 0}{\lim }{V}_{\ast }= \\ =\underset{h\to 0}{\lim }f(c)= \\ =f(x) \\ {F}^{{}^{\prime }}(x)=f(x) \end{gathered}$$