Answer the following questions related to products of polynomials with exponential or trigonometric functions.
(i) State the rule of integration by parts for an indefinite integral.
(ii) Give one clear reason why, when integrating the product of an algebraic polynomial and an exponential function, the polynomial is generally selected as the first function.
(iii) Demonstrate, with a brief argument, that differentiating a polynomial of degree n reduces its degree by one, and illustrate your explanation with a concrete example.