Solution: Preliminary steps The following integrals require

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

ISBN: 9780321570567 2

Solution for problem 51E Chapter 7.4

Calculus: Early Transcendentals | 1st Edition

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

Calculus: Early Transcendentals | 1st Edition

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Problem 51E

Problem 51E

Preliminary steps

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals.

Step-by-Step Solution:

Step 1 of 3

Problem 51E

Answer

We have to evaluate the integral $\int\limits_{\ }^{\ }\frac{3{x}^{2}+4x-6}{{x}^{2}-3x+2}\ dx$

Divide $\frac{3{x}^{2}+4x-6}{{x}^{2}-3x+2}$ we get,

Quotient $3\ \ \ ,reminder\ =13x-12\$

Therefore

$\frac{3{x}^{2}+4x-6}{{x}^{2}-3x+2}=3+\frac{13x-12}{{x}^{2}-3x+2}$

$\Rightarrow{}\int\limits_{\ }^{\ }\frac{3{x}^{2}+4x-6}{{x}^{2}-3x+2}\ dx=\int\limits_{\ }^{\ }(3+\frac{13x-12}{{x}^{2}-3x+2})\ dx$

$=\int\limits_{\ }^{\ }3dx+\int\limits_{\ }^{\ }\frac{13x-12}{{x}^{2}-3x+2}\ dx$ …(1)

Now $we\ evalaute\ \int\limits_{\ }^{\ }\frac{13x-12}{{x}^{2}-3x+2}\ dx$

Take a partial fraction of $\frac{13x-12}{{x}^{2}-3x+2}$

$\frac{13x-12}{{x}^{2}-3x+2}\ =\frac{13x-12}{(x-1)(x-2)}$

$\Rightarrow{}\frac{13x-12}{{x}^{2}-3x+2}\ =\frac{{a}_{0}}{x-1}+\frac{{a}_{1}}{x-2}$

After simplifying we get

$\frac{13x-12}{{x}^{2}-3x+2}\ =\frac{-1}{x-1}+\frac{14}{x-2}$

$=\frac{14}{x-2}-\frac{1}{x-1}$ …(2)

Therefore

$\int\limits_{\ }^{\ }\frac{3{x}^{2}+4x-6}{{x}^{2}-3x+2}\ dx=\int\limits_{\ }^{\ }3dx+\int\limits_{\ }^{\ }\frac{14}{x-2}dx-\int\limits_{\ }^{\ }\frac{1}{x-1}dx$

$=3x+14\ ln\ \left|{x-2}\right|-\left|{x-1}\right|+C$

Therefore

$\int\limits_{\ }^{\ }\frac{3{x}^{2}+4x-6}{{x}^{2}-3x+2}\ dx=3x+14\ ln\ \left|{x-2}\right|-\left|{x-1}\right|+C$

Step 2 of 3

Chapter 7.4, Problem 51E is Solved

Step 3 of 3

Textbook: Calculus: Early Transcendentals

Edition: 1

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

ISBN: 9780321570567

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