Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
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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 8.2.32
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (cos(√x))/(√x) dx
Verified step by step guidance1
Start by identifying a suitable substitution to simplify the integral. Notice that the integrand contains both \( \cos(\sqrt{x}) \) and \( \frac{1}{\sqrt{x}} \), which suggests substituting \( u = \sqrt{x} \).
Express \( u = \sqrt{x} \) in terms of \( x \), so \( u = x^{1/2} \). Then, differentiate both sides with respect to \( x \) to find \( du \): \( du = \frac{1}{2} x^{-1/2} dx = \frac{1}{2\sqrt{x}} dx \).
Solve for \( dx \) in terms of \( du \) and \( x \): \( dx = 2 \sqrt{x} du \). Substitute \( dx \) and \( \sqrt{x} = u \) back into the integral to rewrite it entirely in terms of \( u \).
After substitution, the integral becomes \( \int \frac{\cos(u)}{u} \cdot 2u \, du \). Simplify the expression inside the integral by canceling terms where possible.
Integrate the simplified integral with respect to \( u \), then substitute back \( u = \sqrt{x} \) to express the answer in terms of \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. For example, setting a new variable equal to a function inside the integral can make the integral easier to evaluate.
Recommended video:
07:33
Euler's Method
Integration of Trigonometric Functions
Understanding how to integrate trigonometric functions like cosine is essential. The integral of cos(u) with respect to u is sin(u) plus a constant, which helps in solving integrals involving trigonometric expressions.
Recommended video:
6:04Introduction to Trigonometric Functions
Handling Radicals in Integrals
Integrals involving radicals, such as square roots, often require rewriting the expression in terms of powers or using substitution to simplify the integral. Recognizing how to manipulate radicals is key to evaluating such integrals.
Recommended video:
06:13Limits of Rational Functions with Radicals
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)
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Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2 / (x(ln x - 2)³) dx
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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.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx
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Textbook Question
Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?
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