an anti derivative of f . Then ∫ a f x dx = [F( )] = F ( b ) – F( a ) . Remarks In words, the Theorem tells us that a f x dx = (value of the anti derivative F of f at the upper limit b – value of the same anti derivative at the lower limit a ). This theorem is very useful, because it gives us a method of calculating the definite integral more easily, without calculating the limit of a sum.
(iii) The crucial operation in evaluating a definite integral is that of finding a function whose derivative is equal to the integrand. This strengthens the relationship between differentiation and integration. (iv) In a f x dx , the function f needs to be well defined and continuous in [ a , b ]. For instance, the consideration of definite integral ( – ) x x