Q. 54 Calculate each of the integrals ... [FREE SOLUTION]

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Chapter 5: Q. 54 (page 442)

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$鈭� \frac{e^{x}}{e^{3 x} - 2 e^{2 x}} d x$

Short Answer

Expert verified

The value is $\frac{1}{4} x + \frac{1}{2 e^{x}} + \frac{1}{4} \ln | e^{x} - 2 | + C$

Step by step solution

Step 1. Given Information:

Given integral : $鈭� \frac{e^{x}}{e^{3 x} - 2 e^{2 x}} d x$

We want to calculate each of the integrals.

Step 2. Calculation:

$S i m p l i f y : \frac{e^{x}}{e^{3 x} - 2 e^{2 x}} = \frac{e^{x}}{e^{2 x} (e^{x} - 2)} = \frac{1}{e^{x} (e^{x} - 2)} N o w 鈭� \frac{e^{x}}{e^{3 x} - 2 e^{2 x}} d x = 鈭� \frac{1}{e^{x} (e^{x} - 2)} d x S u b s t i t u t i o n e^{x} = u A l s o e^{x} d x = d u 鈬� d x = \frac{d u}{e^{x}} = \frac{d u}{u} A p p l y I n t e g r a l S u b s t i t u t i o n : 鈭� \frac{1}{u (u - 2)} \frac{d u}{u} = 鈭� \frac{1}{u^{2} (u - 2)} d u U \sin g P a r t i a l F r a c t i o n w e g e t 鈭� \frac{1}{u^{2} (u - 2)} d u = - 鈭� \frac{1}{4 u} d u - 鈭� \frac{1}{2 u^{2}} d u + 鈭� \frac{1}{4 (u - 2)} d u = \frac{1}{4} \ln | u | - (- \frac{1}{2 u}) + \frac{1}{4} \ln | u - 2 | + C S u b s i t u d e b a c k u = e^{x} = \frac{1}{4} \ln | e^{x} | - (- \frac{1}{2 e^{x}}) + \frac{1}{4} \ln | e^{x} - 2 | + C = \frac{1}{4} x + \frac{1}{2 e^{x}} + \frac{1}{4} \ln | e^{x} - 2 | + C$

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91影视

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$鈭� \frac{e^{x}}{e^{3 x} - 2 e^{2 x}} d x$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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91影视

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.鈭�exe3x-2e2xdx

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

Applied Mathematics

Theoretical and Mathematical Physics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Calculate each of the integrals in Exercises 53鈥�56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$鈭� \frac{e^{x}}{e^{3 x} - 2 e^{2 x}} d x$