Consider the function defined for all real

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Consider the function

f(x)=x e^{x^{2}}

defined for all real

x

. A student claims that an antiderivative of

f

is given by

F(x)=\dfrac{1}{2}e^{x^{2}}+C

, where

C

is a constant. Verify this claim and analyse the role of the constant through the following steps. (i) Differentiate

F(x)=\dfrac{1}{2}e^{x^{2}}+C

with respect to

x

and simplify the result. (ii) Compare the derivative obtained in part (i) with

f(x)

. Hence decide whether the student’s claim is correct. (iii) Let

G(x)=\dfrac{1}{2}e^{x^{2}}+7

. Determine

G'(x)

and comment on the relationship between

F

and

G

with respect to their derivatives. (iv) State, with reasons, whether any two antiderivatives of

f

can differ by any function other than a constant.