Calculus: Early Transcendentals
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Publisher:James Stewart
Chapter1: Functions And Models
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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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![### Proving Integration Formulas from the Table of Integrals
The task is to prove one of the following integration formulas from the table of integrals:
1. \[
\int \frac{dx}{x^2 \sqrt{a^2 - x^2}} = -\frac{\sqrt{a^2 - x^2}}{a^2 x} + C
\]
2. \[
\int \frac{x^2}{\sqrt{a^2 - x^2}} \, dx = -\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + C
\]
### Detailed Explanation:
1. **Integral of \(\frac{1}{x^2 \sqrt{a^2 - x^2}}\)**:
- The integral \(\int \frac{dx}{x^2 \sqrt{a^2 - x^2}}\) simplifies to \(-\frac{\sqrt{a^2 - x^2}}{a^2 x} + C\).
- Here, \(a\) is a constant and \(C\) represents the constant of integration.
2. **Integral of \(\frac{x^2}{\sqrt{a^2 - x^2}}\)**:
- The integral \(\int \frac{x^2}{\sqrt{a^2 - x^2}} \, dx\) simplifies to \(-\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + C\).
- In this expression, \(\sin^{-1}\) denotes the inverse sine function (also known as the arcsine function), and \(C\) again represents the constant of integration.
### Step-by-Step Proof Strategy:
1. **For the First Integral**:
- Use trigonometric substitution \( x = a \sin \theta\) or \( x = a \cos \theta\) to simplify the integrand.
- Transform the integral into a more manageable form using the corresponding derivatives and identities.
- Integrate using standard trigonometric or algebraic techniques.
- Simplify the result to achieve the final expression.
2. **For the Second Integral**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff015db4-0da8-42fc-9d60-01a672218c35%2F347ca878-281e-4eea-af7a-16b8032d7b64%2Fjnb3uo_processed.png&w=3840&q=75)
Transcribed Image Text:### Proving Integration Formulas from the Table of Integrals
The task is to prove one of the following integration formulas from the table of integrals:
1. \[
\int \frac{dx}{x^2 \sqrt{a^2 - x^2}} = -\frac{\sqrt{a^2 - x^2}}{a^2 x} + C
\]
2. \[
\int \frac{x^2}{\sqrt{a^2 - x^2}} \, dx = -\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + C
\]
### Detailed Explanation:
1. **Integral of \(\frac{1}{x^2 \sqrt{a^2 - x^2}}\)**:
- The integral \(\int \frac{dx}{x^2 \sqrt{a^2 - x^2}}\) simplifies to \(-\frac{\sqrt{a^2 - x^2}}{a^2 x} + C\).
- Here, \(a\) is a constant and \(C\) represents the constant of integration.
2. **Integral of \(\frac{x^2}{\sqrt{a^2 - x^2}}\)**:
- The integral \(\int \frac{x^2}{\sqrt{a^2 - x^2}} \, dx\) simplifies to \(-\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1} \frac{x}{a} + C\).
- In this expression, \(\sin^{-1}\) denotes the inverse sine function (also known as the arcsine function), and \(C\) again represents the constant of integration.
### Step-by-Step Proof Strategy:
1. **For the First Integral**:
- Use trigonometric substitution \( x = a \sin \theta\) or \( x = a \cos \theta\) to simplify the integrand.
- Transform the integral into a more manageable form using the corresponding derivatives and identities.
- Integrate using standard trigonometric or algebraic techniques.
- Simplify the result to achieve the final expression.
2. **For the Second Integral**
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