calculus - Definition of integration

The derivative of a function is defined by $$ f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}} $$ provided the limit exists. For example for $f(x)=\sin(x)$ we can prove that (see here) $$ f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{\sin(x+\Delta x)-\sin(x)}{\Delta x}}=\cos(x) $$ But for integration there are only a set of formulas that come from the above definition (i.e. with knowing the derivative of a function). Is there a general definition for integration like above definition (maybe an anti-limit!) that acts on a function directly?