Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.1.70

Chapter 8, Problem 8.1.70

70. Different methods Let I=∫(x+2)/(x+4)dx.
b. Evaluate I without performing long division on the integrand.

Verified step by step guidance

Step 1: Recognize that the integrand (x+2)/(x+4) can be simplified using substitution. Let u = x + 4, which implies du = dx.

Step 2: Rewrite the numerator in terms of u. Since u = x + 4, we can express x + 2 as (u - 2). Substitute this into the integrand, yielding ∫(u - 2)/u du.

Step 3: Split the integrand into two simpler fractions: ∫(u/u - 2/u) du = ∫1 du - ∫(2/u) du.

Step 4: Evaluate each term separately. The integral of 1 with respect to u is u, and the integral of 2/u with respect to u is 2ln|u|.

Step 5: Substitute back u = x + 4 into the result to express the solution in terms of x. The final expression will be x + 4 - 2ln|x + 4| + C, where C is the constant of integration.

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Key Concepts

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. In this case, recognizing that the integrand can be simplified using substitution or properties of logarithms can help evaluate the integral without long division. Understanding these techniques is crucial for efficiently solving integrals.

Substitution Method

The substitution method involves changing the variable of integration to simplify the integral. For the given integral, one might let u = x + 4, which transforms the integrand into a more manageable form. This technique is particularly useful when the integrand can be expressed in terms of a single variable.

Properties of Logarithms

Properties of logarithms can be applied when integrating rational functions. Specifically, the integral of a function of the form (a + bx)/(c + dx) can often be expressed in terms of logarithmic functions. Recognizing these properties allows for a quicker evaluation of integrals without resorting to polynomial long division.

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Related Practice

Textbook Question

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

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Textbook Question

Visual proof Let F(x)=∫₀ˣ √(a²−t²) dt. The figure shows that F(x)= area of sector OAB+ area of triangle OBC.

a. Use the figure to prove that

F(x) = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2

b. Conclude that ∫ √(a²−x²) dx = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2 + C.

Textbook Question

{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,

∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C

Graph the following functions and find the area under the curve on the given interval.

f(x) = (9 - x²) ⁻², [0, 3/2]

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Textbook Question

102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:

F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).

Verify the following Laplace transforms, where a is a real number.

106. f(t) = cos(at) → F(s) = s/(s² + a²)

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Textbook Question

7–84. Evaluate the following integrals.

62. ∫ from 0 to π/2 √(1 + cosθ) dθ

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

23. ∫ 1/(25 - x²)^(3/2) dx