Solution: Simple linear factors Evaluate the following

Problem 12E Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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Problem 12E

Simple linear factors

Evaluate the following integrals.

Step-by-Step Solution:

Problem 12E

Step 1</p>

In this problem we have to evaluate $\int\limits_{\ }^{\ }\frac{1}{{t}^{2}\ -\ 9}\ dx$

Consider $\int\limits_{\ }^{\ }\frac{1}{{t}^{2}\ -\ 9}\ dx=\int\limits_{\ }^{\ }\frac{1}{(t\ -\ 3)(t\ +\ 3)}\ dt$ … (1)

Let us solve this by using Partial Fractions.

Let us first see about Partial Fractions.

When a proper rational expression is decomposed into a sum of two or more rational expressions, it is known as Partial Fractions.

Step 2</p>

Consider $\frac{1}{(t\ -\ 3)(t\ +\ 3)}=\frac{A}{(t\ -\ 3)}+\frac{B}{(t\ \ +\ 3)}$ … (2)

Taking LCM in the right hand side we get,

$\frac{1}{(t\ -\ 3)(t\ +\ 3)}=\frac{A(t\ +\ 3)\ +\ B(t\ -\ 3)}{(t\ -\ 3)(t\ +\ 3)}$

Since the denominators of both sides are same, we can equate the numerator.

$1=A(t\ +\ 3)+B(t-3)$

Put $t=\ -3$ we get

$1=A(0)-6B$

$\Rightarrow{}\ -6B=1$

$\Rightarrow{}\ B=\ \frac{-1}{6}$

Put $t=3$ we get

$1=6A+B(0)$

$\Rightarrow{}6A=1$

$\Rightarrow{}\ A=\frac{1}{6}$

Thus (2) becomes,

$\frac{1}{(t\ -\ 3)(t\ +\ 3)}=\frac{1}{6(t\ -\ 3)}-\frac{1}{6(t\ +\ 3)}$

Step 3 of 3

Chapter 7.4, Problem 12E is Solved

Textbook: Calculus: Early Transcendentals

Edition: 1

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

ISBN: 9780321570567

The answer to “Simple linear factors Evaluate the following integrals.” is broken down into a number of easy to follow steps, and 7 words. Calculus: Early Transcendentals was written by Sieva Kozinsky and is associated to the ISBN: 9780321570567. The full step-by-step solution to problem: 12E from chapter: 7.4 was answered by Sieva Kozinsky, our top Calculus solution expert on 03/03/17, 03:45PM. This full solution covers the following key subjects: evaluate, factors, Integrals, Linear, simple. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. Since the solution to 12E from 7.4 chapter was answered, more than 268 students have viewed the full step-by-step answer.