Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ √x e√x dx
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 8.3.12
Evaluate the integrals in Exercises 1–22.
∫ cos³(2x) sin⁵(2x) dx
Verified step by step guidance1
Recognize that the integral involves powers of sine and cosine functions with the same argument, which suggests using trigonometric identities or substitution to simplify the expression.
Express one of the trigonometric functions in terms of the other using the Pythagorean identity. For example, rewrite \(\cos^3(2x)\) as \(\cos(2x) \cdot \cos^2(2x)\) and then use \(\cos^2(2x) = 1 - \sin^2(2x)\).
Rewrite the integral as \(\int \cos(2x) (1 - \sin^2(2x)) \sin^5(2x) \, dx\), which separates the powers of sine and cosine and prepares for substitution.
Use the substitution \(u = \sin(2x)\), then compute \(du = 2 \cos(2x) \, dx\), or equivalently, \(\cos(2x) \, dx = \frac{du}{2}\), to rewrite the integral entirely in terms of \(u\).
Substitute into the integral to get an integral in terms of \(u\): \(\int (1 - u^2) u^5 \frac{du}{2}\), then simplify and integrate the resulting polynomial expression with respect to \(u\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities like power-reduction and product-to-sum formulas help simplify integrals involving powers of sine and cosine. For example, expressing cos³(2x) as cos(2x)·cos²(2x) and then using cos²(θ) = 1 - sin²(θ) can make the integral more manageable.
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Verifying Trig Equations as Identities
Substitution Method
The substitution method involves changing variables to simplify the integral. In this problem, substituting u = sin(2x) or u = cos(2x) can transform the integral into a polynomial form, making it easier to integrate.
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07:33Euler's Method
Integration of Powers of Sine and Cosine
Integrals involving powers of sine and cosine often require reducing the powers step-by-step or using identities to rewrite the integrand. Recognizing patterns and applying reduction formulas is key to solving such integrals efficiently.
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03:53Derivatives of Sine & Cosine
Related Practice
Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ x / (x + √(x² + 2)) dx
Textbook Question
Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)
To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.
∫ log₂ x dx
Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 8 cos^4(2πt) dt
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ tan^(-1)(√y) dy
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(2θ) cos³(2θ) dθ
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