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 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 8.2.22
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-y) cos(y) dy
Verified step by step guidance1
Identify the integral to solve: \(\int e^{-y} \cos(y) \, dy\).
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).
Choose \(u\) and \(dv\) wisely. For this integral, let \(u = \cos(y)\) and \(dv = e^{-y} dy\).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = -\sin(y) dy\), and integrate \(dv\) to get \(v = \int e^{-y} dy = -e^{-y}\).
Apply the integration by parts formula: substitute \(u\), \(v\), \(du\) into \(\int u \, dv = uv - \int v \, du\) and simplify the resulting integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Integration of Exponential Functions
Exponential functions like e^(-y) have straightforward antiderivatives, often involving a constant factor adjustment. Recognizing how to integrate e^(-y) is essential, as it frequently appears in integrals combined with trigonometric functions.
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Integration of Trigonometric Functions
Integrals involving trigonometric functions such as cos(y) often require knowledge of their antiderivatives, like sin(y) for cos(y). When combined with other functions, these integrals may need repeated application of integration techniques.
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Related Practice
Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
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.
∫ (sec t + cot t)² dt
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx
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Textbook Question
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin³(θ) cos(2θ) dθ
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ dx / (8 + 2x²) from 0 to 2