Let be a continuous function on and let and be two antiderivatives of such that and…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

f

be a continuous function on

[a,b]

and let

F(x)

and

G(x)

be two antiderivatives of

f(x)

such that

F'(x)=f(x)

and

G'(x)=f(x)

for all

x

[a,b]

. (a) Prove that

F(x)

and

G(x)

differ only by a constant; that is,

G(x)=F(x)+k

for some constant

k

. (b) Using part (a), establish that

\displaystyle\int_{a}^{b}f(x)\,dx=F(b)-F(a)=G(b)-G(a)

, demonstrating that the definite integral is independent of the choice of antiderivative and of the constant

k

. (c) Verify the result in part (b) for the function

f(x)=3x^{2}-4x+1

[0,2]

by evaluating the definite integral once with the antiderivative

F(x)=x^{3}-2x^{2}+x+7

and again with

G(x)=x^{3}-2x^{2}+x-4