ABSTRACT
The basic rule for finding the constant value of the definite integral b () ∫ a b f(x) dx https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq491.tif"/> is as follows. Given a function f(x) which is continuous on [a,b], find an antiderivative F(x) of f(x), that is, a function which satisfies F ′ ( x ) ≡ f ( x ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq492.tif"/> and then apply the formula () ∫ a b f(x)=F(b) − F(a) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq493.tif"/> Unfortunately, too often this rule cannot be applied, even though every continuous function f(x) has an antiderivative. To make this possibly ambiguous point clear, consider f(x) = x 2 , 1 ≤ x ≤ 2. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq494.tif"/> Suppose we wish to determine the constant value of ∫ 1 2 x 2 dx . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq495.tif"/> Then, () F(x) = ∫ 0 x t 2 dt https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq496.tif"/> 77is an antiderivative of f(x), since F ′ ( x ) ≡ x 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq497.tif"/> by the Fundamental Theorem of Calculus. Unfortunately, (3.3) is of no practical value when applied in (3.2), since it yields ∫ 1 2 x 2 dx = ( ∫ 0 x t 2 dt ) | 1 2 = ∫ 0 2 t 2 dt − ∫ 0 1 t 2 dt = ∫ 1 2 t 2 dt . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq498.tif"/> On the other hand, if one uses the antiderivative () F(x) = x 3 3 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq499.tif"/> then (3.2) yields ∫ 1 2 x 2 dx = x 3 3 | 1 2 = 8 3 − 1 3 = 7 3 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq500.tif"/> and the constant value of the integral has been found.
