Skip to main content
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 8.7.71
Chapter 8, Problem 8.7.71

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)

Verified step by step guidance
1
Step 1: Recognize that the integral ∫[x/(ax + b)] dx can be simplified using substitution. Let u = ax + b, which implies that du = a dx.
Step 2: Rewrite x in terms of u. Since u = ax + b, solving for x gives x = (u - b)/a.
Step 3: Substitute u and du into the integral. Replace x with (u - b)/a and dx with du/a, transforming the integral into ∫[((u - b)/a) / u] * (1/a) du.
Step 4: Simplify the integral. Factor out constants (1/a²) and split the fraction into two terms: ∫[(u/a²) / u] du - ∫[(b/a²) / u] du.
Step 5: Evaluate each term separately. The first term simplifies to ∫(1/a²) du, and the second term simplifies to ∫(-b/(a²u)) du. Use the standard integral formulas for these terms to proceed further.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. Understanding integration techniques, such as substitution, is essential for solving integral problems.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Substitution Method

The substitution method is a technique used in integration to simplify the process of finding an integral. By substituting a part of the integrand with a new variable (often denoted as 'u'), the integral can be transformed into a more manageable form. In the given problem, substituting 'u = ax + b' simplifies the integral, making it easier to evaluate.
Recommended video:
07:33
Euler's Method

Rational Functions

Rational functions are ratios of polynomials, and they often appear in integration problems. The integral of a rational function can sometimes be simplified using algebraic manipulation or substitution. In this case, the function x/(ax + b) is a rational function, and understanding its structure is crucial for applying the appropriate integration techniques.
Recommended video:
6:04
Intro to Rational Functions
Related Practice
Textbook Question

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

88
views
Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

38. ∫ tan⁵θ sec⁴θ dθ

57
views
Textbook Question

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.

82. ∫ (sin⁻¹(ax)) / x² dx, a > 0

40
views
Textbook Question

94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.

For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?

What is its value when it exists?

50
views
Textbook Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

53
views
Textbook Question

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

23. ∫ 1/(25 - x²)^(3/2) dx

46
views