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...
Related questions
Question
Solve using Trig Substitution.
![Certainly! Here is the transcription of the image along with a brief educational explanation:
---
### Integral Transcription and Explanation
The mathematical integral presented is:
\[ \int \frac{x}{x^2 - a^2} \, dx \]
#### Explanation:
This integral represents the area under the curve of the function \(\frac{x}{x^2 - a^2}\) with respect to \(x\).
To solve this integral, a common technique used is substitution. A suitable substitution for this type of integral is \(u = x^2 - a^2\), which simplifies the integral and aids in finding the antiderivative.
##### Steps for Solving:
1. **Substitution:**
Let \(u = x^2 - a^2\). Then, \(du = 2x \, dx \Rightarrow \frac{1}{2} du = x \, dx\).
2. **Rewrite the Integral:**
Substitute \(u\) and \(\frac{1}{2} du\) into the integral:
\[
\int \frac{x}{x^2 - a^2} \, dx = \int \frac{1}{u} \cdot \frac{1}{2} du = \frac{1}{2} \int \frac{1}{u} \, du.
\]
3. **Integrate:**
The integral of \(\frac{1}{u}\) is \(\ln |u| + C\) (where \(C\) is the constant of integration):
\[
\frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln |u| + C.
\]
4. **Substitute Back \(u\):**
Finally, substitute \(u = x^2 - a^2\) back into the equation:
\[
\int \frac{x}{x^2 - a^2} \, dx = \frac{1}{2} \ln |x^2 - a^2| + C.
\]
Thus, the solution to the integral is:
\[
\int \frac{x}{x^2 - a^2} \, dx = \frac{1}{2} \ln |x^2 - a^2| + C.
\]
Understanding](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6a53db3-c060-4093-8251-3e3936495142%2F40fe9d75-0f76-4d30-85c2-141859b8b358%2F9zz0q2i_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription of the image along with a brief educational explanation:
---
### Integral Transcription and Explanation
The mathematical integral presented is:
\[ \int \frac{x}{x^2 - a^2} \, dx \]
#### Explanation:
This integral represents the area under the curve of the function \(\frac{x}{x^2 - a^2}\) with respect to \(x\).
To solve this integral, a common technique used is substitution. A suitable substitution for this type of integral is \(u = x^2 - a^2\), which simplifies the integral and aids in finding the antiderivative.
##### Steps for Solving:
1. **Substitution:**
Let \(u = x^2 - a^2\). Then, \(du = 2x \, dx \Rightarrow \frac{1}{2} du = x \, dx\).
2. **Rewrite the Integral:**
Substitute \(u\) and \(\frac{1}{2} du\) into the integral:
\[
\int \frac{x}{x^2 - a^2} \, dx = \int \frac{1}{u} \cdot \frac{1}{2} du = \frac{1}{2} \int \frac{1}{u} \, du.
\]
3. **Integrate:**
The integral of \(\frac{1}{u}\) is \(\ln |u| + C\) (where \(C\) is the constant of integration):
\[
\frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln |u| + C.
\]
4. **Substitute Back \(u\):**
Finally, substitute \(u = x^2 - a^2\) back into the equation:
\[
\int \frac{x}{x^2 - a^2} \, dx = \frac{1}{2} \ln |x^2 - a^2| + C.
\]
Thus, the solution to the integral is:
\[
\int \frac{x}{x^2 - a^2} \, dx = \frac{1}{2} \ln |x^2 - a^2| + C.
\]
Understanding
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