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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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 8.5.46
Evaluate the integrals in Exercises 39–54.
∫ 1 / ((x¹/³ - 1)√x) dx
(Hint: Let x = u⁶.)
Verified step by step guidance1
Start with the integral \( \int \frac{1}{(x^{1/3} - 1) \sqrt{x}} \, dx \). The hint suggests the substitution \( x = u^6 \), so express \( x \) in terms of \( u \).
Calculate \( dx \) in terms of \( du \) by differentiating \( x = u^6 \), which gives \( dx = 6u^5 \, du \).
Rewrite the expressions inside the integral using the substitution: \( x^{1/3} = (u^6)^{1/3} = u^2 \) and \( \sqrt{x} = \sqrt{u^6} = u^3 \). Substitute these into the integral along with \( dx = 6u^5 \, du \).
Simplify the integral by substituting all parts: the denominator becomes \( (u^2 - 1) u^3 \), and the numerator is replaced by \( 6u^5 \, du \). This will simplify the integral to a rational function in terms of \( u \).
After simplification, set up the integral in terms of \( u \) and proceed to integrate using appropriate techniques such as partial fraction decomposition or algebraic manipulation.
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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 manageable form. By letting x = u⁶, the integral's complicated expressions involving roots and powers become simpler polynomials in u, making integration straightforward.
Recommended video:
07:33
Euler's Method
Handling Radicals and Fractional Exponents
Radicals like √x and fractional exponents such as x^(1/3) can be rewritten as powers with rational exponents. Understanding how to manipulate these expressions is essential for applying substitution and simplifying the integral before integrating.
Recommended video:
Guided course
7:39Introduction to Exponent Rules
Integration of Rational Functions
After substitution, the integral often reduces to a rational function in terms of u. Knowing how to integrate rational functions, possibly by partial fractions or direct power rule application, is crucial to find the antiderivative.
Recommended video:
6:04Intro to Rational Functions
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
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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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Textbook Question
What is the largest value that
∫ from a to b x√(2x - x²) dx
can have for any a and b? Give reasons for your answer.
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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.
∫ x √(x² - 4) dx