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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.3.47
8장, 문제 8.3.47

9–61. Trigonometric integrals Evaluate the following integrals.
47. ∫ (csc⁴x)/(cot²x) dx

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Step 1: Begin by rewriting the integrand in terms of sine and cosine functions. Recall that csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x). Substitute these definitions into the integrand: ∫ (csc⁴x)/(cot²x) dx = ∫ (1/sin⁴x) / (cos²x/sin²x) dx.
Step 2: Simplify the fraction by dividing. This results in ∫ (1/sin⁴x) * (sin²x/cos²x) dx = ∫ (sin²x/sin⁴x) * (1/cos²x) dx. Simplify further to get ∫ (1/sin²x) * (1/cos²x) dx.
Step 3: Recognize that 1/sin²x is equivalent to csc²x and 1/cos²x is equivalent to sec²x. Rewrite the integrand as ∫ csc²x * sec²x dx.
Step 4: Consider substitution or trigonometric identities to simplify the integral further. For example, you might use the identity csc²x = 1 + cot²x to express the integrand in terms of cot(x), or explore substitution methods such as u = cot(x).
Step 5: After substitution or simplification, proceed to integrate the resulting expression. Remember to back-substitute if you used a substitution method, and include the constant of integration (C) in your final answer.

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주요 개념

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, cosecant, and cotangent, are fundamental in calculus, particularly in integration. They relate angles to ratios of sides in right triangles and have periodic properties. Understanding their identities and relationships is crucial for simplifying integrals involving these functions.
추천 영상:
6:04
Introduction to Trigonometric Functions

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In this case, recognizing patterns or using trigonometric identities can simplify the integral. Mastery of these techniques allows for the effective evaluation of integrals that may not be straightforward.
추천 영상:
06:18
Integration by Parts for Definite Integrals

Trigonometric Identities

Trigonometric identities, such as Pythagorean identities and reciprocal identities, are equations involving trigonometric functions that hold true for all values of the variables. These identities can be used to rewrite integrals in a more manageable form, facilitating easier integration. Familiarity with these identities is vital for solving integrals involving trigonometric functions.
추천 영상:
7:17
Verifying Trig Equations as Identities
관련 실천
교과서 질문

9–40. Integration by parts Evaluate the following integrals using integration by parts.

17. ∫ x · 3x dx

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교과서 질문

102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:

F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).

Verify the following Laplace transforms, where a is a real number.

104. f(t) = t → F(s) = 1/s²

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교과서 질문

23-64. Integration Evaluate the following integrals.

32. ∫ (4x - 2)/(x³ - x) dx

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교과서 질문

Choosing an integration strategy Identify a technique of integration for evaluating the following integrals. If necessary, explain how to first simplify the integrand before applying the suggested technique of integration. You do not need to evaluate the integrals.

∫ (1 + tan x) sec²x dx

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교과서 질문

9–61. Trigonometric integrals Evaluate the following integrals.

15. ∫ sin³x cos²x dx

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63. Average Lifetime The average time until a computer chip fails (see Exercise 62) is 0.00005 ∫(from 0 to ∞) t e^(-0.00005t) dt. Find this value.

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