Textbook Question
82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
84. ∫ (from 0 to π) sec²x dx*(Note: Potential improperness at x = π/2)*
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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.R.35
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
35. ∫ x³/√(4x² + 16) dx
Verified step by step guidance1
Step 1: Observe the integral ∫ x³/√(4x² + 16) dx. Notice that the denominator involves a square root of a quadratic expression. This suggests that substitution or trigonometric techniques might be useful.
Step 2: Simplify the quadratic expression inside the square root. Factor out the constant 4 from the term 4x² + 16 to rewrite it as 4(x² + 4). This simplifies the square root to √(4(x² + 4)) = 2√(x² + 4).
Step 3: Perform substitution to simplify the integral. Let u = x² + 4, which implies that du = 2x dx. Rewrite the integral in terms of u: ∫ x³/√(4x² + 16) dx = ∫ x² * x / (2√(u)) dx = ∫ x² / (2√(u)) * (du / 2x).
Step 4: Simplify further using the substitution. Since u = x² + 4, x² = u - 4. Substitute this into the integral: ∫ x² / (2√(u)) * (du / 2x) = ∫ (u - 4) / (2√(u)) * (du / 2x). Cancel out x terms and simplify the integral.
Step 5: Break the integral into simpler parts. After substitution and simplification, you will have two separate integrals to evaluate: one involving u/√(u) and another involving constants. Use integration techniques such as power rule or logarithmic integration to solve each part.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. Each method is suited for different types of integrals, and understanding when to apply each technique is crucial for solving complex integrals effectively.
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Integration by Parts for Definite Integrals
Substitution Method
The substitution method involves changing the variable of integration to simplify the integral. This is particularly useful when the integrand contains a composite function. By substituting a new variable, the integral can often be transformed into a more manageable form, making it easier to evaluate.
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07:33Euler's Method
Rational Functions and Square Roots
Rational functions are ratios of polynomials, and integrals involving square roots often require special techniques. In this case, recognizing the structure of the integrand, such as the presence of √(4x² + 16), can guide the choice of substitution or trigonometric identities to simplify the integral before evaluation.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
15. ∫ (from 1 to 2) (3x⁵ + 48x³ + 3x² + 16)/(x³ + 16x) dx
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Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
71. ∫ (2x² - 4x)/(x² - 4) dx
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
54. ∫ dx/√(9x² - 25), x > 5/3
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
46. ∫ (x³ + 4x² + 12x + 4)/((x² + 4x + 10)²) dx
Textbook Question
95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.
97. ∫ (from 0 to 1) tan(x²) dx; n = 40