Q. 52 Calculate each of the definite i... [FREE SOLUTION]

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Chapter 5: Q. 52 (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.
${鈭�}_{1}^{2} \frac{1}{x (x + 2) (x - 3)} d x$

Short Answer

Expert verified

The value is $\frac{- \ln (2) - 3 \ln (3)}{30}$

Step by step solution

Step 1. Given Information:

Given definite integral : ${鈭�}_{1}^{2} \frac{1}{x (x + 2) (x - 3)} d x$

We want to calculate each of the definite integrals.

Step 2. Calculation:

Use partial fraction we get:

$\frac{1}{x (x + 2) (x - 3)} = \frac{A}{x} + \frac{B}{x + 2} + \frac{C}{x - 3} 1 = A (x^{2} - x - 6) + B (x^{2} - 3 x) + C (x^{2} + 2 x) P lugging 鈥� the 鈥� real 鈥� roots 鈥� of 鈥� the 鈥� denominator : 鈥� 0, 鈥� - 2, 3 F o r d e n o m i n a t o r r o o t 0 : A = - \frac{1}{6} F o r d e n o m i n a t o r r o o t - 2 : B = \frac{1}{10} F o r d e n o m i n a t o r r o o t 3 : C = \frac{1}{15} Use these values we get : \frac{1}{x (x + 2) (x - 3)} = - \frac{1}{6 x} + \frac{1}{10 (x + 2)} + \frac{1}{15 (x - 3)} N o w {鈭�}_{1}^{2} \frac{1}{x (x + 2) (x - 3)} d x = - \frac{1}{6} {鈭�}_{1}^{2} \frac{1}{x} d x + \frac{1}{10} {鈭�}_{1}^{2} \frac{1}{x + 2} d x + \frac{1}{15} {鈭�}_{1}^{2} \frac{1}{x - 3} d x = - \frac{1}{6} {[\ln | x |]}_{1}^{2} + \frac{1}{10} {[\ln | x + 2 |]}_{1}^{2} + \frac{1}{15} {[\ln | x - 3 |]}_{1}^{2} = - \frac{1}{6} [\ln | 2 | - \ln | 1 |] + \frac{1}{10} [\ln | 4 | - \ln | 3 |] + \frac{1}{15} [\ln | - 1 | - \ln | - 2 |] = - \frac{1}{6} \ln (2) + \frac{1}{10} [2 \ln (2) - \ln (3)] - \frac{1}{15} \ln (2) U \sin g \log p r o p e r t i e s w e g e t = \frac{- \ln (2) - 3 \ln (3)}{30}$

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91影视

Calculate each of the definite integrals in Exercises 47鈥�52. Some integrals require partial fractions or polynomial long division, and some do not.
${鈭�}_{1}^{2} \frac{1}{x (x + 2) (x - 3)} d x$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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91影视

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�121x(x+2)(x-3)dx

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

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Geometry

Pure Maths

Logic and Functions

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.
${鈭�}_{1}^{2} \frac{1}{x (x + 2) (x - 3)} d x$