Chapter 5: Q. 48 (page 442)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information:

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

We want to calculate each of the definite integrals.

Step 2. Calculation:

We have

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

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

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 47–52. Some integrals require partial fractions or polynomial long division, and some do not.∫-114(x2-1)x2-4dx

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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