The family f(x) = 1/xp revisited Consider the family of functions f(x) = 1/xp, where p is a real number. For what values of p does the integral exist? What is its value?
Problem 60EThe family revisited Consider the family of functions , where p is a real number. For what values of p does the integral exist What is its valueSolutionStep 1In this problem we have to find for what values of p does the integral exist.We have That is If p = 1 (i.e. p = 1), then this integral becomes Using a new variable b = 1/a, (where b goes to as a goes to 0), this limit becomes log b = , so the integral doesn’t exist. If p 1, then the integral is given by: If (i.e. p < 1), then the above limit exists and equals , whereas if the above limit is infinite (i.e. doesn’t exist). Thus the integral exists if .
