5.8 Remarks on the fundamental theorem of calculus

(i) We write

(ii) A definite integral is a number, namely an area, and the Theorem says that

(iii) It is confusing to write $F(x)=\int_{0}^{x}f(x)\,dx$, so we choose another symbol for the variable of integration.

(iv) The Theorem says that integration is the inverse operation of differentiation. We write

where $C$ is an arbitrary constant. An indefinite integral is a function, which depends upon an arbitrary constant. To check an indefinite integral, differentiate the result.