(i) Give the definition of the Heaviside function H(x).
(ii) Show that H'(x) = S(x), where 8(x) is the Dirac delta function.
(iii) Compute the following integrals
∫x 1√TH (t) dt
∫x -[infinity] sin (╥/2) $(t²-9) dt
where x is a real number. Express your results in terms of the Heaviside function.