Answered: The family f(x) ; revisited Consider…

The family f(x) ; revisited Consider the family of func- tions f(x) 1 where p is a real number. For what values of p does the integral f(x) dx exist? What is its value?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.3: Trigonometric Functions Of Real Numbers

Problem 80E

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Question

The family f(x)
; revisited Consider the family of func-
tions f(x)
1
where p is a real number. For what values of p
does the integral f(x) dx exist? What is its value?

Expert Solution

Step 1

Given $f (x) = \frac{1}{x^{p}}$ , where $p$ is real number.

To find value of $p$ for which $\int_{0}^{1} f (x) d x$ exist.

Solution:

$\begin{array}{rcl} \int_{0}^{1} f (x) d x & = & \int_{0}^{1} \frac{1}{x^{p}} d x \\ = & \int_{0}^{1} x^{- p} d x \\ = & {[\frac{x^{- p + 1}}{- p + 1}]}_{0}^{1} \\ = & \frac{1^{- p + 1}}{- p + 1} - 0 \\ = & \frac{1}{- p + 1} \end{array}$