Q. 48 Calculate each of the definite i... [FREE SOLUTION]

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Chapter 5: Q. 48 (page 442)

Calculate each of the definite integrals in Exercises 47鈥�52. Some integrals require partial fractions or polynomial long division, and some do not.
${鈭�}_{- 1}^{1} \frac{4 (x^{2} - 1)}{x^{2} - 4} d x$

Short Answer

Expert verified

The value is $- \frac{3}{2} \ln (3)$

Step by step solution

Step 1. Given Information:

Given definite integral : ${鈭�}_{- 1}^{1} \frac{4 (x^{2} - 1)}{x^{2} - 4} d x$

We want to calculate each of the definite integrals.

Step 2. Calculation:

We have

${鈭�}_{- 1}^{1} \frac{4 (x^{2} - 1)}{x^{2} - 4} d x = 4 {鈭�}_{- 1}^{1} \frac{x^{2} - 1}{x^{2} - 4} d x U s e l o n g d i v i s i o n m e t h o d w e g e t \frac{x^{2} - 1}{x^{2} - 4} = 1 + \frac{3}{x^{2} - 4} N o w 4 {鈭�}_{- 1}^{1} \frac{x^{2} - 1}{x^{2} - 4} d x = 4 {鈭�}_{- 1}^{1} (1 + \frac{3}{x^{2} - 4}) d x = 4 ({鈭�}_{- 1}^{1} 1 d x + {鈭�}_{- 1}^{1} \frac{3}{x^{2} - 4} d x) S o l v e {鈭�}_{- 1}^{1} \frac{3}{x^{2} - 4} d x = 3 {鈭�}_{- 1}^{1} \frac{1}{x^{2} - 4} d x S u b s t i t u t i o n x = 2 u W h e n x = 1 t h e n u = \frac{1}{2} a n d w h e n x = - 1 t h e n u = - \frac{1}{2} a n d a l s o d x = 2 d u A p p l y I n t e g r a l S u b s t i t u t i o n : 3 {鈭�}_{- 1}^{1} \frac{1}{x^{2} - 4} d x = 3 {鈭�}_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{4 u^{2} - 4} 2 d u = \frac{3}{2} {鈭�}_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{u^{2} - 1} d u = - \frac{3}{2} {鈭�}_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{1 - u^{2}} d u = - \frac{3}{2} {[\frac{\ln |u + 1|}{2} - \frac{\ln |u - 1|}{2}]}_{- \frac{1}{2}}^{\frac{1}{2}} S i m p l i f y : = - \frac{3}{2} [\frac{\ln |\frac{3}{2}|}{2} - \frac{\ln |- \frac{1}{2}|}{2} - (\frac{\ln |\frac{1}{2}|}{2} - \frac{\ln |- \frac{3}{2}|}{2})] U s e \log p r o p e r t i e s : = - \frac{3}{2} (\frac{2 \ln (2) + 2 \ln (\frac{3}{2})}{2}) = - \frac{3}{2} \ln (3)$

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Calculate each of the definite integrals in Exercises 47鈥�52. Some integrals require partial fractions or polynomial long division, and some do not.
${鈭�}_{- 1}^{1} \frac{4 (x^{2} - 1)}{x^{2} - 4} d x$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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Calculate each of the definite integrals in Exercises 47鈥�52. Some integrals require partial fractions or polynomial long division, and some do not.鈭�-114(x2-1)x2-4dx

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Pure Maths

Decision Maths

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Calculate each of the definite integrals in Exercises 47鈥�52. Some integrals require partial fractions or polynomial long division, and some do not.
${鈭�}_{- 1}^{1} \frac{4 (x^{2} - 1)}{x^{2} - 4} d x$