Integral of Rational Function using Partial Fractions

Evaluate the integral $\displaystyle \int \frac{x - 4}{x^2 + 2x - 15} \, dx$ by performing partial fraction decomposition.

Partial fraction decomposition is a method used in calculus to integrate rational functions, that is, functions formed by the ratio of two polynomials. This technique is essential when the denominator polynomial can be factored into simpler linear or quadratic terms. By expressing the integrand as a sum of simpler fractions, each part can often be integrated using basic integral formulas or simpler substitutions.

In the given problem, the function to be integrated is a rational expression where the numerator is a polynomial of lower degree than the denominator. The first step is to factor the denominator into its simplest components. If the denominator is a quadratic that can be factored into linear terms, each factor contributes to a partial fraction. The next step is to rewrite the original integrand as a sum of these simpler fractions. The coefficients in the decomposition are determined by equating the original rational function to the sum of its partial fractions and solving for these coefficients.

Once the partial fraction decomposition is done, the integration process involves separately integrating each partial fraction. This often involves straightforward methods such as recognizing standard integral forms or using simple substitution, like u-substitution. The result is a combined expression that represents the antiderivative of the original rational function. Thus, partial fraction decomposition transforms a complex integral into a series of simpler tasks, which collectively address the integration problem.

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