91Ó°ÊÓ

Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions. When dealing with integrals, this technique is particularly useful for transforming a difficult integrand into a sum of simpler fractions that are easier to integrate. In this exercise, we began with the expression \( \frac{8x + 21}{(x+2)(x+3)} \). The goal was to express it as a sum of two fractions: \( \frac{A}{x+2} + \frac{B}{x+3} \).

• Clear denominators by multiplying both sides by the denominator \((x+2)(x+3)\).
• This results in \(8x + 21 = A(x+3) + B(x+2)\).
• Solve for constants \(A\) and \(B\) by plugging in strategic values for \(x\) (like \(-2\) and \(-3\)).

Substituting these values simplifies the process, eliminating one unknown in each case. Solving gives \(A = 5\) and \(B = 3\), making it possible to proceed with integration.

The integrand is the function or expression inside the integral sign that we aim to integrate. For the given problem, the integrand is \( \frac{8x + 21}{(x+2)(x+3)} \). Analyzing the integrand resets the stage for integrating more complex functions by attempting partial fraction decomposition.

• Identify the integrand: The core expression that needs integration, typically a rational function in problems like this.
• Simplify if possible: Use algebraic techniques like partial fractions to break it into digestible components.

This procedure helps isolate the variables for budding integration and exposes recognizable integral forms that align with standard methods of integration.

Integration by substitution is a technique akin to the reverse of the chain rule from differentiation. However, in this exercise, our main focus is on a simple adaptation through decomposition and direct integration of the components obtained via partial fractions, bypassing the need for more complex substitutions.After sequence separation into simpler fractions like \( \frac{5}{x+2} + \frac{3}{x+3} \), you can integrate directly without substitution.

• This step simplifies the task by allowing straightforward integration of functions with the form \(\frac{k}{x+a}\), requiring no further substitutions.

Such cases are direct logarithmic integrations forming a crucial building block in integration strategies.

Logarithmic integration directly applies when integrating functions of the form \( \frac{k}{x+a} \), where \(k\) is a constant. This is the final step after partial fraction decomposition, making use of an integral identity:

• The integral \( \int \frac{k}{x+a} \, dx = k \ln|x+a| + C \).

Apply this straightforwardly to solve each part of the partial fractions:- For \( \frac{5}{x+2} \), compute \( 5 \ln|x+2| \) evaluated from 1 to 2.- For \( \frac{3}{x+3} \), compute \( 3 \ln|x+3| \) evaluated from 1 to 2.Combining logarithmic integrals provides a neat and efficient closure to your definite integral problem. Simplifying these results involves properties of logarithms such as:- Difference of logs: \( \ln a - \ln b = \ln\frac{a}{b} \).This simplifies calculating and combining separate integrals efficiently into a single expression.