Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ 7cos⁷(t) dt
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 8.3.10
Evaluate the integrals in Exercises 1–22.
∫₀^(π/6) 3cos⁵(3x) dx
Verified step by step guidance1
Recognize that the integral is \( \int_0^{\frac{\pi}{6}} 3 \cos^5(3x) \, dx \). The constant 3 can be factored out: \( 3 \int_0^{\frac{\pi}{6}} \cos^5(3x) \, dx \).
Use the substitution \( u = 3x \), which implies \( du = 3 \, dx \) or \( dx = \frac{du}{3} \). Also, change the limits of integration: when \( x=0 \), \( u=0 \); when \( x=\frac{\pi}{6} \), \( u=\frac{\pi}{2} \).
Rewrite the integral in terms of \( u \): \( 3 \int_0^{\frac{\pi}{6}} \cos^5(3x) \, dx = 3 \int_0^{\frac{\pi}{2}} \cos^5(u) \cdot \frac{du}{3} = \int_0^{\frac{\pi}{2}} \cos^5(u) \, du \).
To integrate \( \cos^5(u) \), express it as \( \cos^4(u) \cos(u) \) and use the identity \( \cos^2(u) = 1 - \sin^2(u) \) to rewrite \( \cos^4(u) = (\cos^2(u))^2 = (1 - \sin^2(u))^2 \).
Use the substitution \( t = \sin(u) \), so \( dt = \cos(u) \, du \). The integral becomes \( \int_0^{\frac{\pi}{2}} \cos^5(u) \, du = \int_0^1 (1 - t^2)^2 \, dt \). Expand the integrand and integrate term-by-term.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Trigonometric Functions
This involves finding the integral of functions involving sine, cosine, or other trig functions. Techniques often include using identities to simplify powers or products of trig functions before integrating.
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Introduction to Trigonometric Functions
Trigonometric Power-Reduction Formulas
Power-reduction formulas express powers of sine or cosine in terms of functions with lower powers or multiple angles, making integration manageable. For example, cos^5(θ) can be rewritten using these formulas to simplify the integral.
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6:04Introduction to Trigonometric Functions
Definite Integration and Limits
Definite integration calculates the exact area under a curve between two bounds. After finding the antiderivative, you evaluate it at the upper and lower limits (here 0 and π/6) and subtract to get the final value.
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Related Practice
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