Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 5 - Integration문제 5.5.22
5장, 문제 5.5.22

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ [ 1/(10𝓍―3) d𝓍

검증된 단계별 안내
1
Step 1: Recognize that the integral ∫ [1/(10𝓍 - 3)] d𝓍 can be solved using a substitution method. Let u = 10𝓍 - 3, which simplifies the denominator.
Step 2: Compute the derivative of u with respect to 𝓍. Since u = 10𝓍 - 3, du/d𝓍 = 10. Rearrange to express d𝓍 in terms of du: d𝓍 = du/10.
Step 3: Substitute u and d𝓍 into the integral. The integral becomes ∫ [1/u] * (du/10), which simplifies to (1/10) ∫ [1/u] du.
Step 4: Recall the standard integral formula ∫ [1/u] du = ln|u| + C, where C is the constant of integration. Apply this formula to the integral.
Step 5: Replace u with the original variable to return to the terms of 𝓍. Since u = 10𝓍 - 3, the solution becomes (1/10) ln|10𝓍 - 3| + C.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
1m
도움이 되었나요?

주요 개념

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation, and it is fundamental in calculus for solving problems related to area under curves and accumulation functions.
추천 영상:
05:04
Introduction to Indefinite Integrals

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex functions or when the integrand can be expressed in a simpler way through a different variable.
추천 영상:
06:35
Changing Geometries

Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. If the derivative of the antiderivative matches the original integrand, the solution is confirmed to be correct. This step is crucial in calculus as it ensures that the integration process has been performed accurately and helps identify any potential errors in the calculations.
추천 영상:
05:02
Determining Differentiability Graphically
관련 실천
교과서 질문

Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.

46
views
교과서 질문

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.                       

                                                                                                                                                                                       

 ∫₀⁵ (𝓍²―9) d𝓍 

74
views
교과서 질문

Areas of regions Find the area of the following regions.                                                                                                                   

                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = x /√(𝓍² ―9) and the 𝓍-axis between and 𝓍 = 4 and 𝓍= 5

28
views
교과서 질문

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ 2𝓍(𝓍² ― 1)⁹⁹ d𝓍

86
views
교과서 질문

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

47
views
교과서 질문

Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)

85
views