Mathematics

Problem

Let

f(x)=\dfrac{\tan^{-1}x}{1+x^{2}}

for all real

x

. A proposed antiderivative is

H(x)=\dfrac{1}{2}\bigl(\tan^{-1}x\bigr)^{2}+C

, where

C

is a constant. Investigate the validity of this proposal and discuss related observations. a. (i) Compute

H'(x)

by differentiating

H(x)

with respect to

x

. b. (ii) Show explicitly whether

H'(x)

equals

f(x)

for every real

x

. c. (iii) If another student writes

K(x)=\dfrac{1}{2}\bigl(\tan^{-1}x\bigr)^{2}-3

, examine

K'(x)

and explain why

H

and

K

represent the same family of antiderivatives. d. (iv) Explain how parts (i)–(iii) illustrate the definition of an indefinite integral.