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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.117
Chapter 5, Problem 5.5.117

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

Verified step by step guidance
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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Root Test
Related Practice
Textbook Question

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


βˆ«β‚ƒβ· (4𝓍 + 6) d𝓍

Textbook Question

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


4

βˆ‘ Ζ’ (1.5 + k) β€’ 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]

k = 1

with n = ________ .

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


βˆ«β‚‹β‚‚β»ΒΉ 𝓍⁻³ d𝓍

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Textbook Question

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

          n                                                                                                                                                                              

    lim   βˆ‘   π“*β‚– (ln 𝓍*β‚–) βˆ†π“β‚– on [1,2]                                                                                                                                                                            

  βˆ† β†’ 0   k=1                                                                                                                                                                                                                      

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Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 βˆ«β‚/₃^ΒΉ/√³ 4/(9𝓍² + 1) d𝓍

Textbook Question

Determine the intervals on which the function g(𝓍) = βˆ«β‚“β° t / (tΒ² + 1) dt  is concave up or concave down.

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