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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.2.59b
Chapter 8, Problem 8.2.59b

59. Two Methods
b. Evaluate ∫(x / √(x + 1)) dx using substitution.

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1
Identify the integral to solve: \(\int \frac{x}{\sqrt{x + 1}} \, dx\).
Choose a substitution to simplify the integral. Let \(u = x + 1\), so that \(du = dx\) and \(x = u - 1\).
Rewrite the integral in terms of \(u\): replace \(x\) with \(u - 1\) and \(dx\) with \(du\), giving \(\int \frac{u - 1}{\sqrt{u}} \, du\).
Split the integral into two simpler integrals: \(\int \frac{u}{\sqrt{u}} \, du - \int \frac{1}{\sqrt{u}} \, du\), which simplifies to \(\int u^{1/2} \, du - \int u^{-1/2} \, du\).
Integrate each term using the power rule for integrals: \(\int u^{n} \, du = \frac{u^{n+1}}{n+1} + C\), then substitute back \(u = x + 1\) to express the answer in terms of \(x\).

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Key Concepts

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

Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves choosing a substitution that transforms the integral into a simpler form, often by letting a part of the integrand equal a new variable. This method is especially useful when the integral contains a composite function.
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Substitution With an Extra Variable

Algebraic Manipulation of Integrands

Before applying substitution, it is often necessary to rewrite the integrand in a form that makes substitution straightforward. This may involve factoring, expanding, or expressing parts of the integrand in terms of the substitution variable. Proper manipulation helps in identifying the correct substitution and simplifies the integration process.
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Completing the Square to Rewrite the Integrand

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial. An indefinite integral represents a family of functions plus a constant of integration, while a definite integral computes the area under a curve between two limits. In substitution, adjusting the limits or adding the constant is important depending on the integral type.
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Definition of the Definite Integral
Related Practice
Textbook Question

2. Give an example of each of the following.

b. A repeated linear factor

Textbook Question

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

Textbook Question

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.

b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].

Textbook Question

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

c. Generalize part (b) and find the average value of the position on the interval [nπ, (n+1)π], for n = 0, 1, 2, ...

Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

71. Let f(x) = √(sin x).

b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)

Textbook Question

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.