Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sinx·cos²x dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.PE.36
Evaluate the integrals in Exercises 33–36.
∫ [1 / √(9 - x²)] dx
Verified step by step guidance1
Recognize that the integral \( \int \frac{1}{\sqrt{9 - x^2}} \, dx \) resembles the standard form \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \), where \( a = 3 \).
Recall the formula for this type of integral: \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C \), where \( C \) is the constant of integration.
Apply the formula by substituting \( a = 3 \) into the expression, giving \( \arcsin\left(\frac{x}{3}\right) + C \).
Write the final integral expression as \( \int \frac{1}{\sqrt{9 - x^2}} \, dx = \arcsin\left(\frac{x}{3}\right) + C \).
Remember to include the constant of integration \( C \) since this is an indefinite integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Functions Involving Square Roots
Integrals containing expressions like √(a² - x²) often require recognizing standard forms or using trigonometric substitution. These integrals relate to inverse trigonometric functions, which simplify the integration process.
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Integrals Involving Natural Logs: Substitution
Inverse Trigonometric Functions
Inverse trigonometric functions such as arcsin, arccos, and arctan arise naturally when integrating functions involving square roots of quadratic expressions. For example, ∫ dx/√(a² - x²) equals arcsin(x/a) + C.
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06:35Derivatives of Other Inverse Trigonometric Functions
Substitution Method in Integration
Substitution is a technique to simplify integrals by changing variables. For integrals like ∫ dx/√(9 - x²), substituting x = 3 sin θ transforms the integral into a simpler trigonometric form, making it easier to evaluate.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ
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Textbook Question
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to ∞ of (x² * e^(−x)) dx
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Textbook Question
Evaluate the integrals in Exercises 33–36.
∫ [1 / (x(9 - x²))] dx
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Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ t dt / √(9 − 4t²)
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Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [x / (x² + 4x + 3)] dx
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