Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Why is e^x selected as u and not e^2x (to get du /2 = e^x dx in the problem attached?
![**Given Information:**
The integral function is
\[
\int \frac{e^{2x}}{e^{2x} + 3e^x + 2} \, dx.
\]
**Calculation:**
Show the integral function as follows:
\[
\int \frac{e^{2x}}{e^{2x} + 3e^x + 2} \, dx \quad \text{(1)}
\]
Consider \( u = e^x \) \quad \text{(2)}
Modify Equation (2):
\[
\ln u = x \quad \text{(3)}
\]
Differentiate both sides of Equation (3):
\[
\frac{1}{u} \, du = dx
\]
Substitute \( u \) for \( e^x \) and \( \frac{1}{u} \, du \) for \( dx \) in Equation (1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2f288b4-9abe-45b3-924a-5abb903bf487%2Ff10b2d08-dfdd-4e84-8c34-1a749f7583bf%2Fs9x8ik_processed.png&w=3840&q=75)
Transcribed Image Text:**Given Information:**
The integral function is
\[
\int \frac{e^{2x}}{e^{2x} + 3e^x + 2} \, dx.
\]
**Calculation:**
Show the integral function as follows:
\[
\int \frac{e^{2x}}{e^{2x} + 3e^x + 2} \, dx \quad \text{(1)}
\]
Consider \( u = e^x \) \quad \text{(2)}
Modify Equation (2):
\[
\ln u = x \quad \text{(3)}
\]
Differentiate both sides of Equation (3):
\[
\frac{1}{u} \, du = dx
\]
Substitute \( u \) for \( e^x \) and \( \frac{1}{u} \, du \) for \( dx \) in Equation (1).
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