교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
19. ∫[0 to π/3] sin⁵x cos⁻²x dx
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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
8장, 문제 8.4.44
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
44. ∫ 1/√(16 + 4x²) dx
검증된 단계별 안내1
Step 1: Recognize that the integral involves a square root of the form √(a² + u²), which suggests using the trigonometric substitution u = a * tan(θ). Here, identify a = 4 and u = 2x, so let 2x = 4 * tan(θ).
Step 2: Substitute u = 2x = 4 * tan(θ). Compute dx by differentiating: dx = d(2x) = 4 * sec²(θ) dθ.
Step 3: Replace √(16 + 4x²) using the trigonometric identity 1 + tan²(θ) = sec²(θ). Substituting, √(16 + 4x²) becomes √(16 * sec²(θ)) = 4 * sec(θ).
Step 4: Rewrite the integral in terms of θ using the substitutions: ∫ 1/√(16 + 4x²) dx = ∫ (1 / (4 * sec(θ))) * (4 * sec²(θ) dθ). Simplify the expression to ∫ sec(θ) dθ.
Step 5: Integrate ∫ sec(θ) dθ using the standard formula for the integral of sec(θ), which is ln|sec(θ) + tan(θ)| + C. Finally, back-substitute θ using the original substitution 2x = 4 * tan(θ) to express the result in terms of x.
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주요 개념
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Trigonometric Substitution
Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of quadratic expressions. By substituting a variable with a trigonometric function, such as sine or tangent, the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that contain expressions like √(a² + x²), a² - x², or x² - a².
추천 영상:
6:04
Introduction to Trigonometric Functions
Identifying the Right Triangle
When using trigonometric substitution, it is essential to visualize the relationship between the variable and the trigonometric function through a right triangle. For example, in the integral ∫ 1/√(16 + 4x²) dx, we can set x = 2tan(θ), which leads to a right triangle where the opposite side is 2x and the adjacent side is 4. This helps in determining the appropriate trigonometric identities to apply.
추천 영상:
07:39Left, Right, & Midpoint Riemann Sums
Integration Techniques
After performing the substitution, the integral often requires the application of various integration techniques, such as basic integration rules or further substitutions. In the case of the integral ∫ 1/√(16 + 4x²) dx, after substituting and simplifying, one may need to integrate a trigonometric function, which can involve recognizing standard integral forms or using integration by parts if necessary.
추천 영상:
06:18Integration by Parts for Definite Integrals
관련 실천
교과서 질문
23-64. Integration Evaluate the following integrals.
62. ∫ 1/[(x + 1)(x² + 2x + 2)²] dx
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교과서 질문
Use a substitution to reduce the following integrals to ∫ ln u du. Then evaluate using the formula for ∫ ln x dx.
7. ∫ (sec²x) · ln(tan x + 2) dx
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교과서 질문
92–98. Evaluate the following integrals.
97. ∫ tan⁻¹(∛x) dx
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교과서 질문
49–52. {Use of Tech} Simpson’s Rule
Apply Simpson’s Rule to the following integrals. It is easiest to obtain the Simpson’s Rule approximations from the Trapezoid Rule approximations, as in Example 8. Make a table similar to Table 8.8 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
51. ∫(from 0 to π) e⁻ᵗ sin(t) dt = ½(e⁻ᵖⁱ + 1)
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교과서 질문
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
8. ∫ sin 3x cos 2x dx
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