Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) 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 8.6.18
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x arctan(x) dx
Verified step by step guidance1
Identify the integral to solve: \(\int x \arctan(x) \, dx\).
Recognize that this integral is a product of two functions: \(x\) and \(\arctan(x)\), which suggests using integration by parts.
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Choose \(u = \arctan(x)\) (since its derivative is simpler) and \(dv = x \, dx\).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = \frac{1}{1 + x^2} \, dx\), and integrate \(dv\) to get \(v = \frac{x^2}{2}\).
Apply the integration by parts formula: write the integral as \(\frac{x^2}{2} \arctan(x) - \int \frac{x^2}{2} \cdot \frac{1}{1 + x^2} \, dx\), then simplify the integrand and prepare to evaluate the remaining 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 used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing u and dv appropriately simplifies the integral, especially when one function becomes simpler upon differentiation.
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Integration by Parts for Definite Integrals
Inverse Trigonometric Functions
Inverse trigonometric functions, like arctan(x), are the inverses of trigonometric functions and have specific derivatives and integrals. Understanding their properties and derivatives, such as d/dx[arctan(x)] = 1/(1+x²), is essential for integrating expressions involving these functions.
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06:35Derivatives of Other Inverse Trigonometric Functions
Using a Table of Integrals
A table of integrals provides formulas for common integrals, saving time and effort. It is useful for recognizing standard forms and applying known results directly, especially when integrals involve special functions or combinations that are difficult to integrate from first principles.
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08:01Integration Using Partial Fractions
Related Practice
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.
∫ arctan x dx
Textbook Question
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4
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Textbook Question
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
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.
∫₁² (8 dx / (x² - 2x + 2))