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
How can I get rid of the highlighted x in this u-substitution problem?
![## Integration by Substitution Example
This example demonstrates how to use substitution for the integration of \( \int \sec^3{x} \tan{x} \, dx \).
### Step-by-Step Solution:
1. Begin with the integral:
\[
\int \sec^3{x} \tan{x} \, dx
\]
2. Introduce a substitution:
Let \( u = \tan{x} \) which implies that \( du = \sec^2{x} \, dx \).
Rearranging for \( dx \), we get:
\[
dx = \frac{1}{\sec^2{x}} \, du
\]
3. Substitute \( u \) and \( dx \) into the integral:
Since \( \sec^3{x} = \sec{x} \cdot \sec^2{x} \), the integral becomes:
\[
\int \frac{\sec^3{x} \cdot u}{\sec^2{x}} \, du
\]
4. Simplify the expression:
\[
\int \sec{x} \cdot u \, du
\]
At this point, there is a remaining \( \sec{x} \) term, despite substituting \( dx \).
5. Identifying the problem:
There is a highlighted text in the image indicating:
*How to get rid of this \( x \)?*
To handle the extra \(\sec{x}\) term in \( \sec{x} \cdot u \, du \), note that \(\sec{x} = \sqrt{u^2 + 1}\) because \(\sec^2{x} = 1 + \tan^2{x}\).
### Summary
To complete the integral, substitute \(\sec{x}\) using the relation to \( u \):
\[
\sec{x} = \sqrt{1 + u^2}
\]
Transforming the integral fully in terms of \( u \):
\[
\int \sqrt{1 + u^2} \, u \, du
\]
The final result requires further simplification or a different technique to integrate the expression completely.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F1cb226ee-3f7c-49d9-bf8a-40cf5f54b102%2Fvyfqum9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Integration by Substitution Example
This example demonstrates how to use substitution for the integration of \( \int \sec^3{x} \tan{x} \, dx \).
### Step-by-Step Solution:
1. Begin with the integral:
\[
\int \sec^3{x} \tan{x} \, dx
\]
2. Introduce a substitution:
Let \( u = \tan{x} \) which implies that \( du = \sec^2{x} \, dx \).
Rearranging for \( dx \), we get:
\[
dx = \frac{1}{\sec^2{x}} \, du
\]
3. Substitute \( u \) and \( dx \) into the integral:
Since \( \sec^3{x} = \sec{x} \cdot \sec^2{x} \), the integral becomes:
\[
\int \frac{\sec^3{x} \cdot u}{\sec^2{x}} \, du
\]
4. Simplify the expression:
\[
\int \sec{x} \cdot u \, du
\]
At this point, there is a remaining \( \sec{x} \) term, despite substituting \( dx \).
5. Identifying the problem:
There is a highlighted text in the image indicating:
*How to get rid of this \( x \)?*
To handle the extra \(\sec{x}\) term in \( \sec{x} \cdot u \, du \), note that \(\sec{x} = \sqrt{u^2 + 1}\) because \(\sec^2{x} = 1 + \tan^2{x}\).
### Summary
To complete the integral, substitute \(\sec{x}\) using the relation to \( u \):
\[
\sec{x} = \sqrt{1 + u^2}
\]
Transforming the integral fully in terms of \( u \):
\[
\int \sqrt{1 + u^2} \, u \, du
\]
The final result requires further simplification or a different technique to integrate the expression completely.
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