Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (x² √(x² + 1))
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 8.2.68
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
Verified step by step guidance1
Identify the parts for integration by parts: let \(u = x^n\) and \(dv = \sin(x) \, dx\).
Compute the derivatives and integrals needed: \(du = n x^{n-1} \, dx\) and \(v = -\cos(x)\) (since \(\frac{d}{dx}(-\cos(x)) = \sin(x)\)).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(\int x^n \sin(x) \, dx = -x^n \cos(x) - \int (-\cos(x)) (n x^{n-1}) \, dx\).
Simplify the integral expression: \(-x^n \cos(x) + n \int x^{n-1} \cos(x) \, dx\).
This establishes the reduction formula: \(\int x^n \sin(x) \, dx = -x^n \cos(x) + n \int x^{n-1} \cos(x) \, dx\).
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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 appropriately is crucial to simplify the integral effectively.
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Integration by Parts for Definite Integrals
Reduction Formula
A reduction formula expresses an integral involving a parameter (like n) in terms of a similar integral with a lower parameter value. It helps solve integrals recursively by breaking them down step-by-step, making complex integrals manageable through repeated application.
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5:59Recursive Formulas
Handling Powers of x in Integration
When integrating expressions like x^n multiplied by trigonometric functions, powers of x are reduced systematically. Differentiating x^n reduces the power by one, which is essential in forming the reduction formula and simplifying the integral through repeated integration by parts.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫₀^π/2 x³ cos 2x dx
Textbook Question
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx
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Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].
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Evaluate the integrals in Exercises 1–22.
∫ cos³(4x) dx
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Evaluate the integrals in Exercises 33–52.
∫ cot⁶(2x) dx