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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 5.5.11
Chapter 5, Problem 5.5.11

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.
∫(𝓍 + 1)ΒΉΒ² d𝓍

Verified step by step guidance
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Step 1: Identify the substitution. Let u = 𝓍 + 1. This substitution simplifies the expression inside the integral.
Step 2: Compute the derivative of u with respect to 𝓍. Since u = 𝓍 + 1, we have du/d𝓍 = 1, or equivalently, du = d𝓍.
Step 3: Rewrite the integral in terms of u. Substituting u = 𝓍 + 1 and du = d𝓍, the integral becomes ∫uΒΉΒ² du.
Step 4: Apply the power rule for integration. The integral of uⁿ with respect to u is (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration.
Step 5: Substitute back u = 𝓍 + 1 into the result to express the solution in terms of 𝓍. This completes the evaluation of the indefinite integral.

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Key Concepts

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

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. By substituting a new variable, often denoted as 'u', for a function of 'x', the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler variable.
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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative gives the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral involves determining the antiderivative of the function, which can often be achieved through various techniques, including substitution.
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Introduction to Indefinite Integrals

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the context of integration, recognizing polynomial forms is crucial, as they can be integrated using straightforward rules. For example, the integral of x^n is (x^(n+1))/(n+1) + C, which simplifies the evaluation of integrals involving polynomials.
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Related Practice
Textbook Question

Average height of a wave The surface of a water wave is described by y = 5 (1 + cos 𝓍) , for ― Ο€ ≀ 𝓍 ≀ Ο€, where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [ ―π , Ο€] .

Textbook Question

Areas of regions Find the area of the region bounded by the graph of Ζ’ and the 𝓍-axis on the given interval.


Ζ’(𝓍) = 𝓍³ ― 1 on [―1, 2]

Textbook Question

Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles.

{Use of Tech} v = 4 √(t +1) (mi/hr) . for 0 ≀ t ≀ 15 ; n = 5     

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Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dz βˆ«ΒΉβ°β‚›α΅’β‚™ β‚‚ dt /(t⁴ + 1)

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Textbook Question

The composite function Ζ’(g(𝓍)) consists of an inner function g and an outer function Ζ’. If an integrand includes Ζ’(g(𝓍)), which function is often a likely choice for a new variable u?

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Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  βˆ« 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)