91Ó°ÊÓ

In calculus, a definite integral is an essential concept used to find the accumulated quantity like area under a curve within a specific interval. The integral of a function between two points, say from 4 to 8 as in the given problem, gives the definite integral. This process involves evaluating an antiderivative at the upper limit and subtracting the value of the antiderivative at the lower limit.
To solve an integral problem like this one, certainly breaking down complex expressions using techniques such as partial fraction decomposition can simplify the process. When faced with the definite integral \( \int_{a}^{b} f(x) \, dx \), it's important to account for the exact limits \(a\) and \(b\) in the calculations.

Factorization is the process of breaking down a complex expression into simpler components, which can be multiplied to give the original expression. This process significantly simplifies integrals involving rational functions, as seen in this exercise.
To accomplish factorization, look for patterns or use formulas that help split polynomials into factors. In this exercise, the expression \( y^2 - 2y - 3 \) is factored into two parts: \( (y - 3)(y + 1) \).
This step transforms a complex fraction into manageable parts. By separating them, we can later employ partial fraction decomposition to make integration practical and straightforward.

The integral evaluation involves solving the problem by finding the antiderivative of the function and using the definite integral limits. Once a rational function is broken down into partial fractions, each separate fraction can be integrated independently.
For the integral \( \int \frac{1}{x} \, dx \) encountered here as part of the decomposition, the antiderivative is \( \ln |x| + C \).
Therefore, given two separate integrals after partial fraction decomposition like \( \int_{4}^{8} \left( \frac{3/4}{y-3} + \frac{1/4}{y+1} \right) \, dy \), solve each piece by substituting the limits of the integral and subtracting the results as outlined in the solution.

Logarithms play a crucial role in solving integrals that involve expressions like \( \int \frac{1}{x} \, dx \). One key property is \( \ln(ab) = \ln(a) + \ln(b) \), which helps combine or separate log terms effectively.
This exercise demonstrates the use of such properties, with logarithmic identities aiding in simplifying the expression further by factoring constants out of the logarithms. For instance, applying \( \frac{1}{4} \ln 9 - \frac{1}{4} \ln 5 \) can be rewritten by these properties into a single logarithm expression: \( \ln \left( \frac{9}{5} \right) \).
Understanding these logarithm properties ensures you can transform your final answer into a neater, condensed form as seen in the solution's conclusion.