How does the fundamental theorem of calculus connect derivatives and integrals?

1 Answer
Sep 11, 2014

The fundamental theorem of calculus shows that differentiation and integration are reverse processes of each other.

Let us look at the statements of the theorem.
(I) $\frac{d}{\mathrm{dx}} {\int}_{a}^{x} f \left(t\right) \mathrm{dx} = f \left(x\right)$
(II) $\int f ' \left(x\right) \mathrm{dx} = f \left(x\right) + C$

As you can see above, (I) shows that integration can be undone by differentiation, and (II) shows that differentiation can be undone by integration (with a loss of the information of C).