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9. Use a trig substitution to show that f steps. dx (1+x²)3/2 = √1+2+C. Be sure to clearly show all

9. Use a trig substitution to show that f steps. dx (1+x²)3/2 = √1+2+C. Be sure to clearly show all

Calculus: Early Transcendentals
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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### Problem 9: Trigonometric Substitution Use a trigonometric substitution to show that: \[ \int \frac{dx}{(1+x^2)^{3/2}} = \frac{x}{\sqrt{1+x^2}} + C. \] Be sure to clearly show all steps. --- **Steps for the Solution:** 1. **Substitution:** Let \( x = \tan(\theta) \). Therefore, \( dx = \sec^2(\theta) d\theta \). 2. **Replacement in the Integral:** Substitute \( x = \tan(\theta) \) into \( (1 + x^2) \): \[ 1 + x^2 = 1 + \tan^2(\theta) = \sec^2(\theta). \] Thus, the integral becomes: \[ \int \frac{\sec^2(\theta) d\theta}{(\sec^2(\theta))^{3/2}}. \] 3. **Simplifying the Integral:** Simplify \( (\sec^2(\theta))^{3/2} \): \[ (\sec^2(\theta))^{3/2} = (\sec(\theta))^3. \] Therefore, the integral becomes: \[ \int \frac{\sec^2(\theta) d\theta}{\sec^3(\theta)} = \int \frac{d\theta}{\sec(\theta)} = \int \cos(\theta) d\theta. \] 4. **Integrate:** \[ \int \cos(\theta) d\theta = \sin(\theta) + C. \] 5. **Back-Substitution:** Recall \( x = \tan(\theta) \). We need to express \(\sin(\theta)\) in terms of \(x\): \[ \sin(\theta) = \frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}} = \frac{x}{\sqrt{1+x^2}}. \] 6. **Final Answer:** \[ \sin(\theta) + C = \frac{x}{\sqrt{1+x^2}} + C. \] Thus, we have shown that: \[ \int \
Transcribed Image Text:### Problem 9: Trigonometric Substitution Use a trigonometric substitution to show that: \[ \int \frac{dx}{(1+x^2)^{3/2}} = \frac{x}{\sqrt{1+x^2}} + C. \] Be sure to clearly show all steps. --- **Steps for the Solution:** 1. **Substitution:** Let \( x = \tan(\theta) \). Therefore, \( dx = \sec^2(\theta) d\theta \). 2. **Replacement in the Integral:** Substitute \( x = \tan(\theta) \) into \( (1 + x^2) \): \[ 1 + x^2 = 1 + \tan^2(\theta) = \sec^2(\theta). \] Thus, the integral becomes: \[ \int \frac{\sec^2(\theta) d\theta}{(\sec^2(\theta))^{3/2}}. \] 3. **Simplifying the Integral:** Simplify \( (\sec^2(\theta))^{3/2} \): \[ (\sec^2(\theta))^{3/2} = (\sec(\theta))^3. \] Therefore, the integral becomes: \[ \int \frac{\sec^2(\theta) d\theta}{\sec^3(\theta)} = \int \frac{d\theta}{\sec(\theta)} = \int \cos(\theta) d\theta. \] 4. **Integrate:** \[ \int \cos(\theta) d\theta = \sin(\theta) + C. \] 5. **Back-Substitution:** Recall \( x = \tan(\theta) \). We need to express \(\sin(\theta)\) in terms of \(x\): \[ \sin(\theta) = \frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}} = \frac{x}{\sqrt{1+x^2}}. \] 6. **Final Answer:** \[ \sin(\theta) + C = \frac{x}{\sqrt{1+x^2}} + C. \] Thus, we have shown that: \[ \int \
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