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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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.PE.6
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² sin(1 − x) dx
Verified step by step guidance1
Identify the parts of the integral for integration by parts. Let \( u = x^2 \) and \( dv = \sin(1 - x) \, dx \).
Compute \( du \) and \( v \): differentiate \( u \) to get \( du = 2x \, dx \), and integrate \( dv \) to find \( v = -\cos(1 - x) \) (remember to consider the chain rule when integrating \( \sin(1 - x) \)).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions for \( u, v, du \) into this formula.
Simplify the resulting integral \( \int v \, du \) and determine if it requires another integration by parts or a simpler method to solve.
Evaluate the remaining integral and combine all parts to express the original integral in terms of \( x \), including the constant of integration \( + C \).
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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 derived from the product rule of 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 simplifies the problem, especially when dealing with products of polynomials and trigonometric functions.
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Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial. Typically, u is chosen as a function that simplifies when differentiated (like polynomials), and dv is chosen as a function that is easy to integrate (like sine or cosine). This choice reduces the complexity of the resulting integral.
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07:51Choosing a Convergence Test
Handling Composite Functions in Integration
When the integrand includes composite functions such as sin(1 - x), recognizing the inner function and applying substitution or careful differentiation is important. Differentiating or integrating these functions requires the chain rule or substitution to correctly evaluate the integral.
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3:48Evaluate Composite Functions - Special Cases
Related Practice
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.
125. ∫ dx / (√x * √(1 + x))
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.
∫ √(2x − x²) dx
Textbook Question
Evaluate the integrals in Exercises 33–36.
∫ [1 / √(9 - x²)] dx
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.
∫ (z + 1) / [z²(z² + 4)] dz
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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)