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Integration Using Partial Fraction Decomposition with Repeated Linear Factors

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Integrate 1x1+1x2+1(x2)2\frac{1}{x - 1} + \frac{1}{x - 2} + \frac{1}{(x - 2)^2} using partial fraction decomposition, acknowledging repeated linear factors.

This problem involves the technique of partial fraction decomposition, a powerful method used to simplify integrands, particularly those involving rational functions. When the integrand is composed of fractions with polynomial denominators, transforming these fractions into simpler components can vastly ease the integration process. In this problem, we encounter the specific situation of repeated linear factors, which adds an extra layer of complexity to the decomposition process. In general, when dealing with repeated linear terms, such as (x2)2(x - 2)^2, they must be accounted for by considering separate terms for each power of these factors. For instance, in (x2)2(x - 2)^2, both 1x2\frac{1}{x - 2} and 1(x2)2\frac{1}{(x - 2)^2} need separate coefficients in the decomposition. Understanding how to systematically set up and solve the corresponding equations to find these coefficients is critical to the successful application of this method. The overarching strategy for problems like these centers around transforming the integrand into a sum of simpler fractions, each of which can be integrated using basic techniques. Handling repeated factors correctly ensures a smooth path to obtaining a solution. This problem serves as an excellent opportunity to practice this integral decomposition strategy, honing the skills necessary for more complex integrals encountered in advanced calculus courses.

Posted by grwgreg 5 days ago

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