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.
∫ (1 - r²)^(5/2) / r⁸ dr
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 8.3.56
Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx
Verified step by step guidance1
Recognize that the integral is \( \int_0^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx \). This is a definite integral with limits from 0 to \( \frac{\pi}{2} \).
Use a trigonometric identity or substitution to simplify the integrand. One useful identity is \( \sin(2x) = 2 \sin(x) \cos(x) \), so \( \sin(x) \cos(x) = \frac{1}{2} \sin(2x) \).
Rewrite the integral using this identity: \( \int_0^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx = \int_0^{\frac{\pi}{2}} \frac{1}{2} \sin(2x) \, dx \).
Factor out the constant \( \frac{1}{2} \) from the integral: \( \frac{1}{2} \int_0^{\frac{\pi}{2}} \sin(2x) \, dx \).
Perform a substitution to integrate \( \sin(2x) \): let \( u = 2x \), then \( du = 2 dx \) or \( dx = \frac{du}{2} \). Change the limits accordingly and integrate \( \sin(u) \) 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.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the product-to-sum identity sin(x)cos(x) = (1/2)sin(2x) simplifies integration by converting products into sums, making the integral easier to evaluate.
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Fundamental Theorem of Calculus
This theorem links differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫ from a to b of f(x) dx = F(b) - F(a). It allows us to evaluate definite integrals by finding antiderivatives and computing their difference at the bounds.
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Related Practice
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 ∞ of (dθ / (1 + e^θ))
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Textbook Question
Finding arc length
Find the length of the curve
y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.
Textbook Question
18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:
a. the y-axis.
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.
∫ 3 sinh(x/2 + ln 5) dx
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Textbook Question
Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫ cos t dt / (1 - cos t)