Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ 7cos⁷(t) dt
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 8.1.6
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.
∫ dx / (x - √x)
Verified step by step guidance1
Start by rewriting the integral to make it easier to work with. Notice that the denominator is \(x - \sqrt{x}\). To simplify, let’s express everything in terms of \(\sqrt{x}\). Set \(t = \sqrt{x}\), which implies \(x = t^2\) and \(dx = 2t \, dt\).
Substitute \(x = t^2\) and \(dx = 2t \, dt\) into the integral. The integral becomes \(\int \frac{2t \, dt}{t^2 - t}\). Simplify the denominator: \(t^2 - t = t(t - 1)\), so the integral is \(\int \frac{2t}{t(t - 1)} \, dt\).
Simplify the integrand by canceling \(t\) in numerator and denominator, resulting in \(\int \frac{2}{t - 1} \, dt\). This is a simpler rational function integral.
Integrate \(\int \frac{2}{t - 1} \, dt\) by recognizing it as a standard form \(\int \frac{1}{u} \, du = \ln|u| + C\). Here, \(u = t - 1\), so the integral is \(2 \ln|t - 1| + C\).
Finally, substitute back \(t = \sqrt{x}\) to express the answer in terms of \(x\). The result is \(2 \ln|\sqrt{x} - 1| + C\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Algebraic Manipulation
Algebraic manipulation involves rewriting expressions to simplify integrals. In this problem, expressing the denominator in terms of a single variable or factoring can make the integral easier to evaluate. For example, substituting √x with a new variable can transform the integral into a more manageable form.
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Substitution Method
The substitution method replaces a complicated expression with a simpler variable to facilitate integration. Here, letting t = √x converts the integral into a rational function in terms of t, allowing easier integration. This technique is essential when dealing with roots or composite functions.
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Integration of Rational Functions
Integrating rational functions often involves partial fraction decomposition or recognizing standard integral forms. After substitution, the integral becomes a rational function in t, which can be integrated by breaking it into simpler fractions or using known integral formulas.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) dx
Textbook Question
Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to 2 of (dx / (1 - x))
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (cos θ + sin 2θ) dθ
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Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₋π^π (1 - cos²(t))^(3/2) dt
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