Textbook Question
Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.4.34
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ 6 dt / (9t² + 1)²
Verified step by step guidance1
Recognize that the integral has the form \(\int \frac{6}{(9t^{2} + 1)^{2}} \, dt\), which suggests a trigonometric substitution because of the quadratic expression inside the denominator.
Rewrite the denominator as \(9t^{2} + 1 = (3t)^{2} + 1^{2}\), which matches the form \(a^{2} + x^{2}\) where \(a = 1\) and \(x = 3t\).
Use the substitution \(3t = \tan(\theta)\), which implies \(t = \frac{1}{3} \tan(\theta)\) and \(dt = \frac{1}{3} \sec^{2}(\theta) \, d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(t\) and \(dt\), and simplify the expression using the identity \(1 + \tan^{2}(\theta) = \sec^{2}(\theta)\).
Evaluate the resulting integral in \(\theta\), then substitute back \(\theta = \arctan(3t)\) to express the answer in terms of \(t\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like a² + x², a² - x², or x² - a² by substituting x with a trigonometric function. This transforms the integral into a trigonometric integral that is often easier to evaluate. For example, substituting t = (1/3)tan(θ) can simplify the denominator 9t² + 1.
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6:04
Introduction to Trigonometric Functions
Integration of Rational Functions
Integrals involving rational functions, especially those with quadratic expressions in the denominator, often require algebraic manipulation or substitution. Recognizing the form and applying appropriate methods like partial fractions or trigonometric substitution helps in evaluating these integrals effectively.
Recommended video:
6:04Intro to Rational Functions
Chain Rule in Reverse (Substitution Method)
The substitution method involves changing variables to simplify an integral, effectively reversing the chain rule from differentiation. By choosing a substitution that simplifies the integral's expression, such as setting u = 9t² + 1, the integral becomes easier to solve.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x√x + 9) dx
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
1
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Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2 ln(z³)) / (16z) dz
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Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx