Skip to main content
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 8.1.69
8장, 문제 8.1.69

69. Different substitutions
b. Evaluate ∫(tan x sec² x) dx using the substitution u=secx.

검증된 단계별 안내
1
Step 1: Identify the substitution given in the problem. Here, we are instructed to use u = sec(x). This substitution will help simplify the integral.
Step 2: Compute the derivative of u with respect to x. Since u = sec(x), we have du/dx = sec(x)tan(x). Therefore, du = sec(x)tan(x) dx.
Step 3: Rewrite the integral ∫(tan(x) sec²(x)) dx using the substitution u = sec(x). Replace sec(x) with u and tan(x) sec(x) dx with du.
Step 4: After substitution, the integral becomes ∫u du because sec²(x) becomes u² and tan(x) sec(x) dx is replaced by du.
Step 5: Integrate ∫u du to find the antiderivative. The result will be expressed in terms of u, and then substitute back u = sec(x) to express the final answer in terms of x.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating the area under a curve represented by a function over a specified interval. Understanding integration is crucial for solving problems involving accumulation and area, and it often requires techniques such as substitution to simplify the integrand.
추천 영상:
06:18
Integration by Parts for Definite Integrals

Substitution Method

The substitution method is a technique used in integration to simplify complex integrals. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. In this case, using the substitution u = sec(x) allows us to express the integral in terms of u, making it easier to evaluate.
추천 영상:
07:33
Euler's Method

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are essential for simplifying expressions and solving integrals involving trigonometric functions. In this problem, recognizing the relationship between tan(x), sec(x), and their derivatives is key to applying the substitution effectively.
추천 영상:
7:17
Verifying Trig Equations as Identities
관련 실천
교과서 질문

23-64. Integration Evaluate the following integrals.

57. ∫ (x³ + 5x)/(x² + 3)² dx

42
views
교과서 질문

37-40. {Use of Tech} Temperature data

Howdy temperature data for Boulder, Colorado; San Francisco, California; Nantucket, Massachusetts; and Duluth, Minnesota, over a 12-hr period on the same day of January are shown in the figure.

Assume these data are taken from a continuous temperature function T(t). The average temperature (in °F) over the 12-hr period is:

T_avg = (1/12) × ∫(0 to 12) T(t) dt

38. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.

46
views
교과서 질문

65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.

67. ∫tan⁴(3y) dy

126
views
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.

28. ∫ ln² x dx

76
views
교과서 질문

54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:

55. ∫ x² cos(5x) dx

70
views
교과서 질문

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

32. ∫ from 0 to 1 x² 2ˣ dx

45
views