We provide a simple proof of the sifting integral as applied to a special-case discontinuous function of the form f(t)=g(t)+ku(t) and illustrate its application with an example involving a simple passive circuit. This special-case value of the sifting integral is either considered to be ambiguous and meaningless, or is completely avoided in the current educational literature. We hope that our discussion can serve the purpose of highlighting the importance of this property of the sifting integral and its applications in electrical engineering.