Answered: 10 dx. x² – 2x Jo x

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.1: Simplifying Fractions

Problem 35WE

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Question

Consider the integral as attached, To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals for the given integral to converge?

Expert Solution

Step 1

Consider the integral $\int_{0}^{3} \frac{10}{x^{2} - 2 x} d x$

To determine the convergence or divergence of the integral,and we need to analyze how many improper integrals are there.

and what must be true of each of these integrals for the given integral to converge.

Step 2

$F i r s t w e w i l l s e e t h e d i c o n t i n u i t i e s o f t h e f u n c t i o n \frac{10}{x^{2} - 2 x} x^{2} - 2 x = 0 = > x = 0 o r x = 2$ $W e c a n w r i t e \int_{0}^{3} \frac{10}{x^{2} - 2 x} d x = \lim_{a \to 0^{+}} \int_{a}^{b} \frac{10}{x^{2} - 2 x} d x + \lim_{c \to 2^{-}} \int_{b}^{c} \frac{10}{x^{2} - 2 x} + \lim_{d \to 2^{-}} \int_{d}^{3} \frac{10}{x^{2} - 2 x} L e t c \in (0, 2) T h e i n t e g r a l m u s t s p l i t i n t h r e e i m p r o p e r i n t e g r a l s t o c o n t a i n o n e l i m i t p e r i n t e g r a l . A s 0 i s t h e l e f t i n t e g r a l l i m i t w e o n l y n e e d t h e r i g h t h a n d l i m i t . T h e d i c o n t i n u i t y a t x = 2 l i e s i n s i d e t h e i n t e r v a l (0, 3) s o w e w i l l c o n s i d e r b o t h t h e l i m i t s .$

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