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Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.
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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 8.3.66
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
Guida verificata passo dopo passo1
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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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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07:33Euler's Method
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
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