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Use trigonometric substitution to find or evaluate the integral. - X 16 dx

Use trigonometric substitution to find or evaluate the integral. - X 16 dx

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.2: Substitution
Problem 1E: Integration by substitution is related to what differentiation method? What type of integrand...
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Use trigonometric substitution to find or evaluate the integral. (Use \( C \) for the constant of integration.) \[ \int \frac{\sqrt{x^2 - 16}}{x} \, dx \] **Explanation:** To solve the integral using trigonometric substitution, you can use the substitution \( x = 4 \sec(\theta) \). This simplifies the integration process by transforming the integrand into a trigonometric function. Follow these steps: 1. **Substitute \( x = 4 \sec(\theta) \):** \[ dx = 4 \sec(\theta) \tan(\theta) \, d\theta \] 2. **Rewrite the integrand using the trigonometric identity:** \[ \sqrt{(4\sec(\theta))^2 - 16} = 4\tan(\theta) \] 3. **Substitute these into the integral:** \[ \int \frac{4\tan(\theta)}{4 \sec(\theta)} \cdot 4 \sec(\theta)\tan(\theta) \, d\theta \] 4. **Simplify the expression:** \[ \int 4\tan^2(\theta) \, d\theta \] 5. **Use the trigonometric identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \) to simplify the integral:** \[ \int 4(\sec^2(\theta) - 1) \, d\theta = 4 \int (\sec^2(\theta) - 1) \, d\theta \] 6. **Integrate \( \sec^2(\theta) - 1 \):** \[ 4 \left( \int \sec^2(\theta) \, d\theta - \int d\theta \right) = 4 \left( \tan(\theta) - \theta \right) + C \] 7. **Substitute back using \( \theta = \sec^{-1} \left( \frac{x}{4} \right) \) and \( \tan(\theta) = \sqrt{\sec^2(\theta)- 1} = \sqrt{\left(\frac{x}{4}\
Transcribed Image Text:Use trigonometric substitution to find or evaluate the integral. (Use \( C \) for the constant of integration.) \[ \int \frac{\sqrt{x^2 - 16}}{x} \, dx \] **Explanation:** To solve the integral using trigonometric substitution, you can use the substitution \( x = 4 \sec(\theta) \). This simplifies the integration process by transforming the integrand into a trigonometric function. Follow these steps: 1. **Substitute \( x = 4 \sec(\theta) \):** \[ dx = 4 \sec(\theta) \tan(\theta) \, d\theta \] 2. **Rewrite the integrand using the trigonometric identity:** \[ \sqrt{(4\sec(\theta))^2 - 16} = 4\tan(\theta) \] 3. **Substitute these into the integral:** \[ \int \frac{4\tan(\theta)}{4 \sec(\theta)} \cdot 4 \sec(\theta)\tan(\theta) \, d\theta \] 4. **Simplify the expression:** \[ \int 4\tan^2(\theta) \, d\theta \] 5. **Use the trigonometric identity \( \tan^2(\theta) = \sec^2(\theta) - 1 \) to simplify the integral:** \[ \int 4(\sec^2(\theta) - 1) \, d\theta = 4 \int (\sec^2(\theta) - 1) \, d\theta \] 6. **Integrate \( \sec^2(\theta) - 1 \):** \[ 4 \left( \int \sec^2(\theta) \, d\theta - \int d\theta \right) = 4 \left( \tan(\theta) - \theta \right) + C \] 7. **Substitute back using \( \theta = \sec^{-1} \left( \frac{x}{4} \right) \) and \( \tan(\theta) = \sqrt{\sec^2(\theta)- 1} = \sqrt{\left(\frac{x}{4}\
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