91Ó°ÊÓ

Integration is a mathematical concept used to find the area under a curve or function over a specified interval. In this context, we utilize partial fraction decomposition to simplify the integration process. The idea is to break down a more complex rational function into simpler components that are easier to integrate.
In this exercise, the rational function \( \frac{x-7}{x^2-x-12} \) is expressed in a decomposed form. This allows us to integrate each term individually. The integral is transformed into a manageable sum of simpler fractions, specifically \( \int \left( \frac{-3}{x-4} + \frac{4}{x+3} \right) \ dx \).
By integrating each piece separately, we obtain specific results:

• \( \int \frac{-3}{x-4} \, dx = -3 \ln |x-4| + C_1 \)
• \( \int \frac{4}{x+3} \, dx = 4 \ln |x+3| + C_2 \)

These individual integrals are then combined into a single expression: \[ -3 \ln |x-4| + 4 \ln |x+3| + C \], where \( C \) is the constant of integration encompassing both \( C_1 \) and \( C_2 \). This approach simplifies the integration of complex rational expressions efficiently.

Quadratic factorization is a fundamental step in partial fraction decomposition. It involves expressing a quadratic expression as the product of two linear factors.
For the given expression \(x^2 - x - 12\), factorization helps us split into \((x-4)(x+3)\). This transformation is crucial because it allows separating the rational expression into parts.
The factorization process may utilize the quadratic formula or trial and error. Here's a quick breakdown:

• The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), which gives the solutions to \(ax^2 + bx + c = 0\).
• For \(x^2 - x - 12\), this results in roots \(x=4\) and \(x=-3\), thus factorizing into \((x-4)(x+3)\).

By expressing the original quadratic denominator in terms of its factors, the integration task is turned into a set of simpler problems. Hence, factoring is not just an algebraic manipulation but a strategic step to simplify the complex rational function into more accessible fractions.

Coefficient matching is an essential technique to determine the unknown constants when working with partial fraction decomposition. After expressing the function as a sum of simpler fractions, these constants need to be identified to proceed with integration.
For the expression \(\frac{x-7}{(x-4)(x+3)} = \frac{A}{x-4} + \frac{B}{x+3}\), we need to find \(A\) and \(B\).
We start by multiplying through by the original denominator \((x-4)(x+3)\), clearing the fractions:\[ x - 7 = A(x+3) + B(x-4) \]Next, expand and rearrange:\[ x - 7 = Ax + 3A + Bx - 4B \]Now, combine like terms:\[ x - 7 = (A+B)x + (3A-4B) \]From this expression, we equate coefficients on both sides of the equation:

• \( A + B = 1 \)
• \( 3A - 4B = -7 \)

Solving this simple system of equations algebraically, we find \(A = -3\) and \(B = 4\).
This essential step ensures that the partial fraction accurately represents the original expression, allowing us to integrate effectively.