Textbook Question
41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.4.26
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx
Verified step by step guidance1
Step 1: Recognize that the integral involves a square root of the form √(x² - a²), which suggests using a trigonometric substitution. Specifically, let x = sec(θ), where sec²(θ) - 1 = tan²(θ). This substitution simplifies the square root.
Step 2: Substitute x = sec(θ) into the integral. Compute dx = sec(θ)tan(θ)dθ and replace √(x² - 1) with √(sec²(θ) - 1) = tan(θ). The integral becomes ∫ √(x² - 1)/x dx = ∫ tan(θ)/sec(θ) * sec(θ)tan(θ)dθ.
Step 3: Simplify the integrand. The expression tan(θ)/sec(θ) * sec(θ)tan(θ) simplifies to tan²(θ)dθ. The integral now becomes ∫ tan²(θ)dθ.
Step 4: Use the trigonometric identity tan²(θ) = sec²(θ) - 1 to rewrite the integral. The integral becomes ∫ (sec²(θ) - 1)dθ, which can be split into two separate integrals: ∫ sec²(θ)dθ - ∫ 1dθ.
Step 5: Evaluate the two integrals separately. ∫ sec²(θ)dθ = tan(θ), and ∫ 1dθ = θ. After integration, convert back to the original variable x using the substitution x = sec(θ), and apply the given limits of integration √2 to √2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as x = sec(θ) or x = sin(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(x² - a²), √(a² - x²), or √(x² + a²).
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Integration Techniques
Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and partial fractions. Understanding these techniques is essential for solving complex integrals, as they provide strategies to break down the integral into simpler parts. Mastery of these methods allows students to tackle a wide range of integrals, including those that arise from trigonometric substitutions.
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Definite Integrals
Definite integrals represent the accumulation of quantities, calculated over a specific interval [a, b]. They are denoted as ∫[a to b] f(x) dx and yield a numerical value that corresponds to the area under the curve of the function f(x) between the limits a and b. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which connects differentiation and integration.
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Related Practice
Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt
1
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Textbook Question
7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
39. ∫ x²/(100 - x²)^(3/2) dx
Textbook Question
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals
Textbook Question
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
70. ∫ cos(x)cos(2x) dx