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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.4.26
Capitolo 8, Problema 8.4.26

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx

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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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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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Introduction to Trigonometric Functions

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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Integration by Parts for Definite Integrals

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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Definition of the Definite Integral
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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.

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7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

39. ∫ x²/(100 - x²)^(3/2) dx

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76. Apparent discrepancy

Three different computer algebra systems give the following results:

∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).

Explain how all three can be correct.

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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

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7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

54. ∫ y⁴/(1 + y²) dy

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9–40. Integration by parts Evaluate the following integrals using integration by parts.

36. ∫ from 0 to ln2 x eˣ dx

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