Fundamental theorem of calculus

Abstract

In this chapter we study functions of the form

$$F\left( x \right) = \int\limits_a^x {f\left( t \right)} {\rm{ d}}t$$

called indefinite integrals of f. If f. is continuous, then F is an antiderivative of f, see Theorem 9.13. Indefinite integrals allow an easy computation of definite integrals as follows,

$$\int\limits_a^b {f\left( x \right)} {\rm{ d}}x = F\left( b \right) - F\left( a \right),$$

see Theorem 9.15. These two theorems are known jointly as the fundamental theorem of calculus. An application to physics is given in Sect. 9.2.