Two methods Evaluate for x > 1 in two ways: using partial

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567 2

Solution for problem 71E Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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Problem 71E

Two methods Evaluate for x > 1 in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

Step-by-Step Solution:

SOLUTION

Here we need to calculate the value of $\int\limits_{\ }^{\ }\frac{dx}{{x}^{2}-1}$ using partial fraction method and also using trigonometric substitution method

Step 1

Using partial fraction method

$\int\limits_{\ }^{\ }\frac{dx}{{x}^{2}-1}$ = $\int\limits_{\ }^{\ }\frac{dx}{{(x}^{\ }+1)(x-1)}$

Therefore $\frac{1}{{(x}^{\ }+1)(x-1)}=\frac{a}{x+1}+\frac{b}{x-1}$

We get $1=a(x-1)+b(x+1)$

On solving this we get $a=\frac{-1}{2}\ and\ b=\frac{1}{2}$

Therefore we write $\frac{1}{{(x}^{\ }+1)(x-1)}=\frac{a}{x+1}+\frac{b}{x-1}=\frac{-1}{2(x+1)}+\frac{1}{2(x-1)}$

$\int\limits_{\ }^{\ }\frac{dx}{{x}^{2}-1}$ = $\int\limits_{\ }^{\ }\frac{dx}{{(x}^{\ }+1)(x-1)}=\frac{1}{2}\{\int\limits_{\ }^{\ }\frac{1}{x-1}dx-\int\limits_{\ }^{\ }\frac{1}{x+1}dx\}$

$\Rightarrow{}\ \ \frac{1}{2}\{ln(|x-1|)-ln(|x+1|)\}$ +c

Step 2 of 2

Chapter 7.4, Problem 71E is Solved

Textbook: Calculus: Early Transcendentals

Edition: 1

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

ISBN: 9780321570567

The answer to “Two methods Evaluate for x > 1 in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.” is broken down into a number of easy to follow steps, and 21 words. This full solution covers the following key subjects: answers, evaluate, fractions, methods, partial. This expansive textbook survival guide covers 112 chapters, and 5248 solutions. The full step-by-step solution to problem: 71E from chapter: 7.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. Since the solution to 71E from 7.4 chapter was answered, more than 258 students have viewed the full step-by-step answer.