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Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 8, Problema 8.3.68
Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx
Guida verificata passo dopo passo1
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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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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7:17
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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Percorso guidato
06:18Integration by Parts for Definite Integrals
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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Percorso guidato
6:04Introduction to Trigonometric Functions
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