교과서 질문
7–84. Evaluate the following integrals.
59. ∫ 1/(x⁴ + x²) 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 8.3.51
9–61. Trigonometric integrals Evaluate the following integrals.
51. ∫ (csc²x + csc⁴x) dx
검증된 단계별 안내1
Step 1: Recognize that the integral involves trigonometric functions csc²x and csc⁴x. Break the integral into two separate parts: ∫ csc²x dx + ∫ csc⁴x dx.
Step 2: Recall the standard integral formula for csc²x: ∫ csc²x dx = -cot(x) + C, where C is the constant of integration.
Step 3: For ∫ csc⁴x dx, rewrite csc⁴x as (csc²x)². Use the identity csc²x = 1 + cot²x to express csc⁴x in terms of cot²x.
Step 4: Substitute csc²x = 1 + cot²x into (csc²x)² to get csc⁴x = (1 + cot²x)². Expand this expression to obtain 1 + 2cot²x + cot⁴x.
Step 5: Break ∫ (1 + 2cot²x + cot⁴x) dx into separate integrals: ∫ 1 dx + ∫ 2cot²x dx + ∫ cot⁴x dx. Solve each integral using appropriate techniques, such as substitution for cot²x and cot⁴x, and combine the results.
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주요 개념
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Trigonometric Functions
Trigonometric functions, such as sine, cosine, and cosecant, are fundamental in calculus, particularly in integration. The cosecant function, csc(x), is defined as the reciprocal of the sine function, csc(x) = 1/sin(x). Understanding these functions is crucial for evaluating integrals involving trigonometric identities.
추천 영상:
6:04
Introduction to Trigonometric Functions
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and recognizing patterns in integrals. For the integral ∫ (csc²x + csc⁴x) dx, recognizing the forms of csc²x and csc⁴x can help simplify the integration process.
추천 영상:
06:18Integration by Parts for Definite Integrals
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. These identities, such as the Pythagorean identities and reciprocal identities, can simplify integrals. For example, knowing that csc²x = 1 + cot²x can help in breaking down the integral into more manageable parts.
추천 영상:
7:17Verifying Trig Equations as Identities
관련 실천
교과서 질문
Let f(x) = (4x³ + x² + 4x + 2) / (x² + 1). Use long division to show that f(x) = 4x + 1 + 1 / (x² + 1) and use this result to evaluate ∫f(x) dx.
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교과서 질문
1. On which derivative rule is integration by parts based?
83
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교과서 질문
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
9. 4/(x⁵ - 5x³ + 4x)
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교과서 질문
River flow rates
The following figure shows the discharge rate r(t) of the Snoqualmie River near Carnation, Washington, starting on a February day when the air temperature was rising. The variable t is the number of hours after midnight, r(t) is given in millions of cubic feet per hour, and ∫(0 to 24) r(t) dt equals the total amount of water that flows by the town of Carnation over a 24-hour period. Estimate ∫(0 to 24) r(t) dt using the Trapezoidal Rule and Simpson's Rule with the following values of n.
n = 6
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교과서 질문
7–64. Integration review Evaluate the following integrals.
28. ∫ (3x + 1) / √(4 - x²) dx
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