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 dx) / (25 + 4x²)
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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 8.3.66
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
Verified step by step guidance1
Rewrite the integrand \( \frac{\sin^3(x)}{\cos^4(x)} \) by expressing \( \sin^3(x) \) as \( \sin(x) \cdot \sin^2(x) \). This gives \( \int \frac{\sin(x) \cdot \sin^2(x)}{\cos^4(x)} \, dx \).
Use the Pythagorean identity \( \sin^2(x) = 1 - \cos^2(x) \) to rewrite \( \sin^2(x) \) in terms of \( \cos(x) \). Substitute this into the integral to get \( \int \frac{\sin(x) (1 - \cos^2(x))}{\cos^4(x)} \, dx \).
Make the substitution \( u = \cos(x) \), which implies \( du = -\sin(x) \, dx \) or equivalently \( -du = \sin(x) \, dx \). Replace \( \sin(x) \, dx \) in the integral with \( -du \).
Rewrite the integral entirely in terms of \( u \): \( \int \frac{\sin(x)(1 - u^2)}{u^4} \, dx = - \int \frac{1 - u^2}{u^4} \, du \).
Split the integral into simpler terms: \( - \int \left( u^{-4} - u^{-2} \right) du \). Then integrate each term separately using the power rule for integrals.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities and Manipulations
Understanding how to rewrite powers of sine and cosine using identities or algebraic manipulation is essential. For example, expressing sin³(x) as sin(x)·sin²(x) and then using sin²(x) = 1 - cos²(x) helps simplify the integral into a more manageable form.
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Verifying Trig Equations as Identities
Substitution Method
The substitution method involves changing variables to simplify the integral. In this case, substituting u = cos(x) transforms the integral into a rational function of u, making it easier to integrate by reducing trigonometric complexity.
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Integration of Rational Functions
After substitution, the integral often becomes a rational function, which requires techniques like polynomial division or partial fraction decomposition. Mastery of these methods allows for straightforward integration of the resulting algebraic expression.
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6:04Intro to Rational Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx
Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ ln(x + x²) dx
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 √(2x - x^2) dx
Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
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.
∫ (sec t + cot t)² dt