Antiderivative Using Partial Fraction Decomposition

Find the antiderivative of $\frac{x^2 + 9}{(x^2 - 1)(x^2 + 4)}$ using partial fraction decomposition.

This problem involves finding the antiderivative of a rational function by using partial fraction decomposition, a technique used to simplify the integration of complex rational expressions. Partial fraction decomposition is a method where a complex fraction is expressed as a sum of simpler fractions. This approach is particularly useful when dealing with rational functions where the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

In this problem, the decomposition will help break down the rational function into simpler forms that are easier to integrate individually. To effectively solve this problem, it is essential to understand how to decompose the fraction correctly, which involves determining the correct form of partial fractions including linear and possibly repeated or irreducible quadratic factors. Once the rational function is decomposed, each fraction can typically be integrated using basic antiderivative formulas or through straightforward substitutions.

Mastery in identifying the structure of the factors in the denominator and applying the right partial fraction decomposition plays a crucial role in solving this problem successfully. This problem not only reinforces the concept of integration through decomposition but also touches on strategic thinking required for tackling integrals that appear complex at first glance.

By simplifying the problem using partial fractions, it becomes more manageable and provides insight into the broader strategy of integration, which is crucial when encountering various forms of integrals in calculus. Understanding the principles of decomposing fractions and applying correct integrative techniques is a valuable skillset within many mathematical and applied contexts.

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