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
![**Finding the Indefinite Integral**
The problem given is to find the indefinite integral of the following expression:
\[ \int e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) \, dx \]
To solve this problem, we can use substitution.
1. Let \( u = \frac{1}{x} \).
2. Therefore, \( du = -\frac{1}{x^2} \, dx \) which means \( dx = -x^2 \, du \).
Substituting these into the integral, we get:
\[
\int e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) \, dx = \int e^u \, du
\]
3. Now integrate \( e^u \) with respect to \( u \):
\[
\int e^u \, du = e^u + C
\]
4. Finally, substitute \( u \) back in terms of \( x \):
\[
e^u + C = e^{\frac{1}{x}} + C
\]
Therefore, the solution to the given integral is:
\[
\int e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2} \right) \, dx = e^{\frac{1}{x}} + C
\]
Where \( C \) is the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F239df334-5b03-4bc0-9001-d398335a6cd3%2F7ea300f8-3ea0-4161-acd7-f5ba17a7eeff%2Fr4crqzd_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Indefinite Integral**
The problem given is to find the indefinite integral of the following expression:
\[ \int e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) \, dx \]
To solve this problem, we can use substitution.
1. Let \( u = \frac{1}{x} \).
2. Therefore, \( du = -\frac{1}{x^2} \, dx \) which means \( dx = -x^2 \, du \).
Substituting these into the integral, we get:
\[
\int e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2}\right) \, dx = \int e^u \, du
\]
3. Now integrate \( e^u \) with respect to \( u \):
\[
\int e^u \, du = e^u + C
\]
4. Finally, substitute \( u \) back in terms of \( x \):
\[
e^u + C = e^{\frac{1}{x}} + C
\]
Therefore, the solution to the given integral is:
\[
\int e^{\frac{1}{x}} \cdot \left(-\frac{1}{x^2} \right) \, dx = e^{\frac{1}{x}} + C
\]
Where \( C \) is the constant of integration.
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