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I know that the derivative of $f(x)$ must be $$f'(x)=\lim\limits_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ We can use this formula to derive the derivatives of some functions like $\sin(x)$. Is there such a formula for an integral? In other words, is there a set formula for computing: $$\int f(x) \ dx$$ Thanks in advance for your answers!

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there is a formula for definite integrals: That is: Suppose $f: [a,b] \to \mathbb{R}$, then

$$ \int\limits_{[a,b]} f = \lim_{\Delta x_i \to 0} \sum_{ i=0 }^n f( \xi_i) \Delta x_i$$

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