Section 39: The fundamental theorem of integral calculusand the evaluation of definite integrals

We now establish the fundamental connection between definite integrals and

antiderivatives (indefinite integrals). Let a continuous function f(t) be

defined for a0/t0/b and let x be any point inside the interval. Then if we regard a definite integral as a function of its upper limit, which we write as x ,

a function F (x ) may be defined as

F (x)g x

f (t) dt,

where t is a dummy variable.