Antiderivative Using Partial Fractions

Find the anti-derivative of $\frac{x - 9}{x^2 + 3x - 10}$ using partial fractions.

In this problem, we are tasked with finding the antiderivative of a rational function using the method of partial fractions. This technique is particularly useful when dealing with integrals of functions that can be expressed as a ratio of polynomials. The core idea behind partial fractions is to decompose a complex rational function into simpler fractions that are more manageable to integrate individually. By breaking the original function into smaller parts, each with a simpler denominator, we can apply basic integration techniques that might not be feasible with the original format.

The strategy involves factoring the denominator polynomial completely, whenever possible, into its irreducible components. Then, we express the rational function as a sum of simpler fractions that correspond to each of these factors. For example, a quadratic factor $x^2 + bx + c$ would result in a partial fraction decomposition involving terms such as $(Ax + B)$ which are easier to integrate. Understanding how to set up and solve these decomposition equations is crucial, as it directly influences the integration success.

Once the function is expressed in partial form, integration involves handling each term independently. The key advantage here is that these terms are often in a form where basic calculus techniques, like natural logarithmic integration for linear terms, can be applied directly. Therefore, mastering the setup and algebraic manipulation in the partial fraction stage is crucial for successfully integrating complex rational functions.

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