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

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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.

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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.

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