Textbook Question
7–84. Evaluate the following integrals.
7. ∫ from 0 to π/2 [sin θ / (1 + cos² θ)] dθ
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.3.26
9–61. Trigonometric integrals Evaluate the following integrals.
26. ∫ sin³x cos³ᐟ²x dx
Verified step by step guidance1
Rewrite the integral using trigonometric identities. Specifically, use the identity for powers of sine and cosine: \( \sin^3(x) = \sin(x) \cdot \sin^2(x) \) and \( \sin^2(x) = 1 - \cos^2(x) \). This transforms the integral into \( \int \sin(x) (1 - \cos^2(x)) \cos^{3/2}(x) \, dx \).
Perform a substitution to simplify the integral. Let \( u = \cos(x) \), which implies \( du = -\sin(x) \, dx \). Substitute these into the integral, replacing \( \sin(x) \, dx \) with \( -du \) and \( \cos(x) \) with \( u \). The integral becomes \( -\int (1 - u^2) u^{3/2} \, du \).
Expand the integrand \( (1 - u^2) u^{3/2} \) by distributing \( u^{3/2} \). This gives \( -\int (u^{3/2} - u^{7/2}) \, du \).
Integrate each term separately. Use the power rule for integration, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), to find the antiderivative of \( u^{3/2} \) and \( u^{7/2} \).
After finding the antiderivative, substitute back \( u = \cos(x) \) to return to the original variable. Simplify the expression to complete the solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are essential for simplifying integrals involving trigonometric functions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. Understanding these identities helps in transforming complex integrals into simpler forms that are easier to evaluate.
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Integration Techniques
Integration techniques are methods used to find the integral of a function. For trigonometric integrals, techniques such as substitution, integration by parts, and the use of trigonometric identities are often employed. In the case of products of sine and cosine functions, using identities to express the integrand in a more manageable form can significantly simplify the integration process.
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Power Reduction Formulas
Power reduction formulas are specific trigonometric identities that allow us to express powers of sine and cosine in terms of first-degree functions. For example, sin²x can be rewritten using the identity sin²x = (1 - cos(2x))/2. These formulas are particularly useful when integrating higher powers of sine and cosine, as they reduce the degree of the functions, making the integral easier to solve.
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05:58Intro to Power Series
Related Practice
Textbook Question
49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.
52. ∫ from 0 to π/2 of cos⁶x dx
Textbook Question
Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.
∫ (5x² + 18x + 20) / [(2x + 3)(x² + 4x + 8)] dx
Textbook Question
58–61. {Use of Tech} Using Simpson's Rule Approximate the following integrals using Simpson's Rule. Experiment with values of n to ensure the error is less than 10⁻³.
60. ∫(from 0 to π) ln(2 + cos x) dx = π ln((2 + √3)/2)
Textbook Question
23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.
24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals
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Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
43. ∫ tan³(4x) dx