Consider the integrand

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

h(x)=e^{2x}-5\cos x

. A one-parameter family of functions is given by

H_k(x)=\tfrac12 e^{2x}-5\sin x+kx,

where

k

is a real constant. (a) Compute

H_k'(x)

in terms of

k

. (b) Determine the value of

k

(if any) for which

H_k'(x)=h(x)

for every real

x

. Show all algebraic steps leading to your conclusion. (c) Using the value of

k

found in part (b), verify explicitly that

H_k

is an antiderivative of

h

. (d) Explain how your result illustrates both the non-uniqueness of antiderivatives and the restriction imposed on

k

in this particular family.