91Ó°ÊÓ

Partial Fraction Decomposition is a technique used to express a complex rational expression as a sum of simpler fractions. This helps in simplifying the process of integration, especially when dealing with polynomial denominators. In the given exercise, the fraction \( \frac{x+2}{x(x-3)(x+1)} \) is decomposed into partial fractions in order to separate it into more manageable parts. This fraction can be represented as \( \frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+1} \).
To determine the coefficients \(A\), \(B\), and \(C\), we multiply through by the denominator \(x(x-3)(x+1)\), set the resulting expression equal to \(x+2\), and solve for the unknowns by substituting suitable values for \(x\). This method simplifies the integration process by breaking down a complex problem into parts that can be integrated using basic techniques.

Factoring polynomials is a crucial step when working with polynomial expressions, particularly for simplifying fractions in partial fraction decomposition.
This involves breaking down a polynomial into its constituent factors, which are simpler polynomials that, when multiplied together, give the original polynomial. In this exercise, the polynomial denominator \( x^3 - 2x^2 - 3x \) was factored by first extracting the common factor \( x \), leading to \( x(x^2 - 2x - 3) \).

• The quadratic \(x^2 - 2x - 3\) is then further factored into \((x-3)(x+1)\) by finding numbers that multiply to \(-3\) and add to \(-2\).

Factoring simplifies the integrand and reveals the distinct linear factors necessary for partial fraction decomposition.

Integrals are fundamental to calculus and represent the area under a curve or the accumulated quantity of a function over an interval. By applying polynomial factoring and partial fraction decomposition, the task of integrating a complex rational function becomes manageable.

• Once the expression is decomposed, each partial fraction is integrated individually.
• In this case, each fraction leads to a natural logarithm function due to the linear denominators.

This process not only translates a difficult integration problem into simpler parts but also builds a systematic approach to evaluate integrals involving polynomial functions.

Logarithmic Integration comes into play when integrating fractions with a linear denominator, resulting in expressions that involve logarithms. After decomposing the original problem into partial fractions, each term such as \( \int \frac{A}{x} \, dx \) can be integrated using the natural logarithm rule.
For example, integrating \( \frac{A}{x} \) results in \( A\ln|x| \), and similar for other terms involving \(x-3\) or \(x+1\).

• In each case, the integration of \( \frac{1}{u} \, du \) yields \( \ln|u| + C \), where \(u\) represents the linear expression in the denominator.
• This results in a straightforward solution formed by the sum of these logarithm terms.

The final result combines all these logarithmic expressions to provide a complete, integrated solution to the original exercise.