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.
∫ (x² / (x² + 1)) dx
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.3.68
Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx
Verified step by step guidance1
Rewrite the integrand to express it in terms of sine and cosine functions. Recall that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), so the integral becomes \(\int \frac{\cot(x)}{\cos^{2}(x)} \, dx = \int \frac{\cos(x)}{\sin(x) \cos^{2}(x)} \, dx\).
Simplify the integrand by canceling one \(\cos(x)\) term in the numerator and denominator, resulting in \(\int \frac{1}{\sin(x) \cos(x)} \, dx\).
Consider a substitution to simplify the integral. One useful substitution is to let \(u = \sin(x)\), which implies \(du = \cos(x) \, dx\). Rearranging, we get \(dx = \frac{du}{\cos(x)}\).
Substitute \(u\) and \(dx\) back into the integral. The integral becomes \(\int \frac{1}{u \cos(x)} \cdot \frac{du}{\cos(x)} = \int \frac{1}{u \cos^{2}(x)} \, du\). Since this still contains \(\cos(x)\), consider an alternative substitution or rewrite the integrand differently.
Alternatively, rewrite the original integrand as \(\cot(x) \sec^{2}(x)\) because \(\frac{1}{\cos^{2}(x)} = \sec^{2}(x)\). Then, use the substitution \(u = \sin(x)\), so \(du = \cos(x) \, dx\). Express \(\cot(x) \sec^{2}(x) \, dx\) in terms of \(u\) and \(du\) to evaluate the integral.
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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 hold true for all values within their domains. They help simplify integrals by rewriting expressions in more manageable forms, such as expressing cotangent as cos(x)/sin(x) or using Pythagorean identities to transform powers of sine and cosine.
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Verifying Trig Equations as Identities
Integration Techniques
Integration techniques include methods like substitution, integration by parts, and rewriting integrals to simpler forms. For integrals involving trigonometric functions, substitution is often used by identifying a function and its derivative within the integrand to simplify the integral into a basic form.
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Handling Rational Trigonometric Functions
Rational trigonometric functions are ratios of trigonometric expressions, such as cot(x)/cos²(x). Understanding how to manipulate these ratios, often by expressing all terms in sine and cosine, is essential to simplify the integral and apply substitution or other integration methods effectively.
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Related Practice
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.
∫₁² (8 dx / (x² - 2x + 2))
Textbook Question
Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ e√x / √x dx
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.
∫ (tan θ + 3 / sin θ) dθ
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / x^1.001
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