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 8.5.50
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x⁶(x⁵ + 4)) dx
Verified step by step guidance1
Start by rewriting the integral to clearly see the expression: \(\int \frac{1}{x^{6}(x^{5} + 4)} \, dx\).
Consider a substitution to simplify the integral. Notice that \(x^{5}\) appears inside the parentheses, so let \(u = x^{5} + 4\).
Differentiate \(u\) with respect to \(x\) to find \(du\): \(du = 5x^{4} \, dx\), which implies \(dx = \frac{du}{5x^{4}}\).
Rewrite the integral in terms of \(u\) and \(x\): substitute \(x^{5} + 4\) with \(u\) and \(dx\) with \(\frac{du}{5x^{4}}\). Also, express the remaining powers of \(x\) in terms of \(u\) if possible.
Simplify the integral after substitution and look for further algebraic manipulation or partial fraction decomposition to integrate with respect to \(u\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Rational Functions
This involves integrating functions expressed as the ratio of two polynomials. Techniques often include algebraic manipulation, partial fraction decomposition, or substitution to simplify the integral into manageable parts.
Recommended video:
6:04
Intro to Rational Functions
Partial Fraction Decomposition
A method used to break down complex rational expressions into simpler fractions that are easier to integrate. It requires factoring the denominator and expressing the integrand as a sum of simpler rational terms.
Recommended video:
10:07Partial Fraction Decomposition: Distinct Linear Factors
Substitution Method
A technique where a part of the integrand is replaced with a new variable to simplify the integral. It is especially useful when the integral contains a composite function or when the derivative of a function appears within the integrand.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x√x + 9) dx
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
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
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
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