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 8.6.22

Chapter 8, Problem 8.6.22

7–84. Evaluate the following integrals.
22. ∫ [1 / ((x - a)(x - b))] dx, where a ≠ b

Verified step by step guidance

Recognize that the integral involves a rational function with two distinct linear factors in the denominator, (x - a) and (x - b). This suggests using partial fraction decomposition to simplify the integrand.

Express the integrand as a sum of partial fractions: \( \frac{1}{(x - a)(x - b)} = \frac{A}{x - a} + \frac{B}{x - b} \), where A and B are constants to be determined.

Multiply through by \((x - a)(x - b)\) to eliminate the denominators, resulting in \( 1 = A(x - b) + B(x - a) \). Expand and collect terms to solve for A and B.

Substitute the values of A and B back into the partial fraction decomposition, rewriting the integrand as \( \frac{A}{x - a} + \frac{B}{x - b} \). Then, split the integral into two separate integrals: \( \int \frac{A}{x - a} dx + \int \frac{B}{x - b} dx \).

Evaluate each integral using the formula \( \int \frac{1}{x - c} dx = \ln|x - c| + C \), where C is the constant of integration. Combine the results to obtain the final expression for the integral.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

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 and partial fractions, is essential for evaluating integrals.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to break down a complex rational function into simpler fractions that are easier to integrate. This technique is particularly useful when dealing with integrals of rational functions, as it allows for the integration of each simpler fraction separately. In the given integral, the expression 1 / ((x - a)(x - b)) can be decomposed into simpler fractions to facilitate integration.

Definite vs. Indefinite Integrals

Definite integrals calculate the area under a curve between two specific limits, while indefinite integrals represent a family of functions and include a constant of integration. The integral in the question is an indefinite integral, meaning it will yield a general antiderivative of the function. Understanding the distinction between these two types of integrals is crucial for correctly interpreting the results of integration.

Recommended video:

05:43

Definition of the Definite Integral

Related Practice

Textbook Question

7–64. Integration review Evaluate the following integrals.

53. ∫ eˣ sec(eˣ + 1) dx

views

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

24. ∫ dt / √(1 + 4eᵗ)

views

Textbook Question

What are the two general ways in which an improper integral may occur?

views

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

45. ∫ sec²x tan¹ᐟ²x dx

128

views

Textbook Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.