91Ó°ÊÓ

Partial fraction decomposition is a technique that helps in simplifying complex fractions into simpler ones, making them easier to integrate. This is particularly useful when you have a rational expression, like the one in our exercise, where the denominator is a factored polynomial. Understanding the basics of partial fraction decomposition involves breaking down a fraction into a sum of fractions.

To apply this technique, the denominator must be fully factored. For example, with the integral \( \frac{1}{(x+2)(x+4)} \), we can express it as \( \frac{A}{x+2} + \frac{B}{x+4} \). The goal is to determine the values of \( A \) and \( B \) that satisfy the condition when the fractions are combined back.

Here's how you typically go about it:

• Multiply both sides by the original denominator to get rid of the fraction.
• Expand the right side and collect like terms.
• Equate coefficients of corresponding powers of \( x \) and solve the resulting system of equations.

In our exercise, once we equated and solved the system, we found \( A = \frac{1}{2} \) and \( B = -\frac{1}{2} \). This allows us to rewrite the original integral into something much simpler to integrate.

Quadratic expression factorization is crucial when dealing with integrals where the denominator is a quadratic polynomial. Factoring transforms the quadratic expression into a product of simpler factors, which can then be handled separately. In the given problem, the quadratic expression was \(-x^2 - 12x + 8\).

We can factor it by first rewriting as \(-1(x^2 + 12x - 8)\). The next step is to find factors of the constant term \(-8\) that add up to the linear coefficient, which is \(12\). After examining possible combinations, we realize it factors into \(-(x+2)(x+4)\).

Key points for factorization:

• Rewriting the quadratic in standard form \(ax^2 + bx + c\).
• Searching for two numbers that multiply to \(ac\) and sum to \(b\).
• Breaking the middle term if necessary and factoring by grouping.

Understanding this allows the computation of integrals to become more manageable, as seen in our problem, leading to a simpler expression ready for partial fraction decomposition.

Logarithmic properties become quite handy, especially when dealing with integrals of the form \(\frac{1}{x}\), which integrate to logarithmic functions. After integrating, we simplify further using properties of logarithms, which are very powerful for manipulating and simplifying expressions.

Consider the integrated expression: \[\ln|x+2| - \ln|x+4| + C\].

The logarithmic property \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) allows us to combine the two logarithms into \[\ln \left| \frac{x+2}{x+4} \right| + C\].

Benefits of understanding logarithmic properties:

• Turns subtraction of logs into division, simplifying expressions.
• Consolidates expressions, making them neater and often easier to interpret.
• Helps in solving equations where logarithmic terms are present.

Grasping these logarithmic properties helps in correctly simplifying the final solution, essential for achieving and presenting the most reduced form of our integrated result.