Textbook Question
Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.
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 8.1.8
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.
∫ (2 ln(z³)) / (16z) dz
Verified step by step guidance1
Start by simplifying the integrand. Use the logarithm power rule: \(\ln(z^3) = 3 \ln(z)\), so rewrite the integral as \(\int \frac{2 \cdot 3 \ln(z)}{16z} \, dz\).
Simplify the constants in the integrand: \(\frac{2 \cdot 3}{16} = \frac{6}{16} = \frac{3}{8}\). The integral becomes \(\int \frac{3}{8} \cdot \frac{\ln(z)}{z} \, dz\).
Factor out the constant \(\frac{3}{8}\) from the integral: \(\frac{3}{8} \int \frac{\ln(z)}{z} \, dz\).
Recognize that the integral \(\int \frac{\ln(z)}{z} \, dz\) can be solved using substitution. Let \(u = \ln(z)\), then \(du = \frac{1}{z} dz\), which matches the integrand's differential part.
Rewrite the integral in terms of \(u\): \(\frac{3}{8} \int u \, du\). Then integrate \(u\) with respect to \(u\) using the power rule for integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Understanding logarithm properties, such as ln(a^b) = b ln(a), allows simplification of expressions involving logarithms. This is essential for rewriting ln(z³) as 3 ln(z), making the integral easier to handle.
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05:36
Change of Base Property
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. Recognizing parts of the integrand as derivatives of a function helps to choose an appropriate substitution, streamlining the integration process.
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04:27Substitution With an Extra Variable
Basic Integration Rules
Familiarity with basic integration formulas, such as ∫ ln(x)/x dx or power rule integrals, is crucial. These rules help evaluate integrals after simplification or substitution, enabling the calculation of antiderivatives efficiently.
Recommended video:
06:07Basic Rules for Definite Integrals
Related Practice
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.
∫ 6 dt / (9t² + 1)²
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ x² sin(x) dx
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]
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Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)
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Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx