Question

. Improper Integrals. Say that f is integrable on [a,[infinity]) if

limb→[infinity] R b a f(x)dx =: R [infinity] a f(x)dx exists. (a) For which real
values of p does R [infinity] (logx)^p/x dx exist? (b) Show that R [infinity] 0
sinx

Answer 1

The question refers to improper integrals, focusing on convergence and the determination of the range of values for which these integrals are defined. While specific convergence criteria such as the Comparison Test or the p-Test can be applied, details provided in the question are insufficient to solve part (b) precisely.

Step-by-step explanation:

The question poses a problem concerning improper integrals, a concept in calculus dealing with the integration of functions over an unbounded interval. Specifically, part (a) asks for the range of values p such that the improper integral of (log(x))^p/x from a to infinity exists. This involves determining the convergence of the integral, which is subject to the Comparison Test or the p-Test for convergence of improper integrals.

For part (b), without the full question text, we can't provide a specific answer. However, if the question is about the existence of the integral of sin(x) from 0 to infinity, it is known that this integral does not converge to a finite number, but it does exhibit a type of convergence known as the Cauchy Principal Value. In the context of probability density functions and their associated probabilities mentioned in the extended context, the concept of area under the curve is essential, as it represents the probability of events within a certain range.