Explanation Given: The integral: I = ∫ 2 ∞ d x ( x + 3 ) ( x + 1 ) 2 Formula used: I = ∫ a b f ( x ) d x = [ F ( x ) ] a b = F ( b ) − F ( a ) I = ∫ a b f ( x ) d x = ∫ a c 1 f ( x ) d x + ∫ c 1 c 2 f ( x ) d x + ... + ∫ c n − 1 b f ( x ) d x , Where a < c 1 < c 2 < ... < c n − 1 < b ln 1 = 0 Calculation: The integral we have I = ∫ 2 ∞ d x ( x + 3 ) ( x + 1 ) 2 Assume, 1 ( x + 3 ) ( x + 1 ) 2 = A x + 3 + B x + 1 + C ( x + 1 ) 2 Now multiplying both sides by ( x + 3 ) ( x + 1 ) 2 , we get 1= A ( x + 1 ) 2 + B ( x + 3 ) ( x + 1 ) + C ( x + 3 ) ( 1 ) For x = − 1 , we get 1= A × 0 + B × 0 + C ( − 1 + 3 ) ⇒ 1 = 2 C ⇒ C = 1 2 For x = − 3 , we get 1= A × ( − 3 + 1 ) 2 + B × 0 + C × 0 ⇒ 1 = 4 A ⇒ A = 1 4 Compare the coefficient of x 2 , we get 0= A + B ⇒ B = − A = − 1 4 ∴ A = 1 4 , B = − 1 4 , C = 1 2 ∴ 1 ( x + 3 ) ( x + 1 ) 2 = 1 4 ( x + 3 ) + − 1 4 ( x + 1 ) + 1 2 ( x + 1 ) 2 (1) I = ∫ 2 ∞ d x ( x + 3 ) ( x + 1 ) 2 = ∫ 2 ∞ [ 1 4 ( x + 3 ) + − 1 4 ( x + 1 ) + 1 2 ( x + 1 ) 2 ] d x = ∫ 2 ∞ [ 1 4 ( x + 3 ) ] d x + ∫ 2 ∞ [ − 1 4 ( x + 1 )

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Concept explainers

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

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Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

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Question

Chapter 8.7, Problem 44E

To determine

The value of the integral

$I = \int_{2}^{\infty} \frac{d x}{(x + 3) {(x + 1)}^{2}}$

Whether the given integral is convergent or divergent.

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Chapter 8.7, Problem 43E

Chapter 8.7, Problem 45E

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Chapter 8 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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The value of the integral I = ∫ 2 ∞ d x ( x + 3 ) ( x + 1 ) 2 Whether the given integral is convergent or divergent.

Solution Summary: The author explains the formula used to determine the value of the integral, which is convergent or divergent.

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The value of the integral

$I = \int_{2}^{\infty} \frac{d x}{(x + 3) {(x + 1)}^{2}}$

Whether the given integral is convergent or divergent.

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Chapter 8 Solutions

The value of the integral I = ∫ 2 ∞ d x ( x + 3 ) ( x + 1 ) 2 Whether the given integral is convergent or divergent.

Solution Summary: The author explains the formula used to determine the value of the integral, which is convergent or divergent.

Concept explainers

Videos

The value of the integral I=∫2∞dx(x+3)(x+1)2 Whether the given integral is convergent or divergent.

Want to see the full answer?

Chapter 8 Solutions

The value of the integral

$I = \int_{2}^{\infty} \frac{d x}{(x + 3) {(x + 1)}^{2}}$

Whether the given integral is convergent or divergent.