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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 5.5.117
5장, 문제 5.5.117

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    
                                                                                                                                                                    
  ∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)  

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Step 1: Begin by making the substitution u = √(1 + x). This substitution simplifies the square root expressions in the denominator. Compute the derivative of u with respect to x: du/dx = 1 / (2√(1 + x)), or equivalently dx = 2u du.
Step 2: Rewrite the integral in terms of u. Substituting u = √(1 + x) and dx = 2u du, the integral becomes ∫ (2u du) / [√1 + u].
Step 3: Simplify the integral further. Factor out constants if necessary and focus on simplifying the denominator √1 + u.
Step 4: If the resulting integral is still complex, consider a second substitution to simplify further. For example, let v = √(1 + u), and compute dv in terms of du. Substitute this into the integral.
Step 5: After performing the second substitution, simplify the integral and proceed with integration techniques such as basic antiderivatives or partial fractions, depending on the resulting expression.

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주요 개념

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Substitution Method in Integration

The substitution method is a technique used in calculus to simplify the process of integration. It involves replacing a variable with another variable that simplifies the integral, making it easier to solve. This is particularly useful when dealing with complex functions or compositions of functions, as it can transform the integral into a more manageable form.
추천 영상:
07:33
Euler's Method

Multiple Substitutions

Multiple substitutions refer to the use of more than one substitution in the process of solving an integral. This technique is often necessary when the integral involves nested functions or when a single substitution does not sufficiently simplify the expression. By strategically choosing substitutions, one can progressively simplify the integral until it can be easily evaluated.
추천 영상:
04:27
Substitution With an Extra Variable

Understanding Square Roots in Integrals

Square roots in integrals can complicate the integration process, as they often require careful manipulation to simplify. Recognizing how to handle expressions involving square roots, such as using trigonometric or algebraic identities, is crucial. In the context of the given integral, understanding how to express the square root in terms of a new variable can facilitate the substitution process.
추천 영상:
07:15
Root Test
관련 실천
교과서 질문

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.


∫₃⁷ (4𝓍 + 6) d𝓍

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교과서 질문

Identifying Riemann sums Fill in the blanks with an interval and a value of n.


4

∑ ƒ (1.5 + k) • 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]

k = 1

with n = ________ .

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교과서 질문

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.                                

          n                                                                                                                                                                              

    lim   ∑   𝓍*ₖ (ln 𝓍*ₖ) ∆𝓍ₖ on [1,2]                                                                                                                                                                            

  ∆ → 0   k=1                                                                                                                                                                                                                      

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교과서 질문

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

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교과서 질문

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₂^π/² 5 sin θ dθ

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교과서 질문

Determine the intervals on which the function g(𝓍) = ∫ₓ⁰ t / (t² + 1) dt  is concave up or concave down.

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