For a non-negative integer n, define the function
a. In(x)=∫xnexdx.
Consider the family {In} as n varies.
(1) Describe a systematic strategy that relates In(x) to In−1(x) without carrying out the entire antiderivative. In your description, identify which term you would differentiate and which you would leave unchanged, and explain why this choice lowers the exponent on x. Do not compute any integrals.
(2) Based on the strategy you described in part (a), write the general algebraic shape of the relationship you expect between In(x) and In−1(x). Your answer should make clear how polynomial factors in n and powers of x are arranged but need not include exact coefficients.
(3) Explain how mathematical induction, together with the relationship discussed in parts (a) and (b), could be used to obtain an explicit closed-form expression for In(x) valid for all integers n≥0. No computation of the closed form is required; focus on the logical steps of the inductive argument.