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Partial fraction decomposition is a technique used to break down a complex rational expression into simpler fractions that are easier to integrate or work with. Imagine it like peeling an onion: you separate the layers (fractions) to handle each one individually.### Factor the DenominatorTo start partial fraction decomposition, we factor the denominator of the given rational expression. In our example, the expression \[\frac{x^3 + 7x^2 + 9x + 2}{x(x^2 + 3x + 2)}\]has the denominator, \[x(x^2 + 3x + 2),\] which simplifies to \[x(x + 1)(x + 2).\] This process reveals the distinct elements that we need to consider in the decomposition.### Set Up the DecompositionEach factor in the denominator becomes a term in our partial fractions. Our goal is to rewrite the integrand as:\[\frac{A}{x} + \frac{B}{x+1} + \frac{C}{x+2},\]where \(A\), \(B\), and \(C\) are constants we need to find. This setup allows us to handle each fraction separately and simplifies integration.### Solve for the ConstantsBy clearing the denominator through multiplication, a system of equations emerges that allows us to solve for these constants \(A\), \(B\), and \(C\). With our example, matching coefficients gives us \(A = 1\), \(B = 2\), and \(C = 3\). These constants will help rewrite the original complex fraction into simpler, integrable parts.

The definite integral is a concept that allows you to calculate the area under a curve within a set interval. In simpler terms, it measures the accumulation of quantities, such as area, within these bounds.### Applying Definite IntegralsIn our scenario, we need to find the definite integral of each partial fraction over the interval from 1 to 2. By rewriting the problem with constant-filled fractions:\[\int_{1}^{2} \frac{1}{x} \, dx + \int_{1}^{2} \frac{2}{x+1} \, dx + \int_{1}^{2} \frac{3}{x+2} \, dx,\]each fraction is integrated separately, reflecting how the definite integral accumulates value across a specified interval.### Calculating Each IntegralFor each fraction, we use known integral results:

• \( \int_{1}^{2} \frac{1}{x} \, dx \) results in \( \ln(2) \),
• \( \int_{1}^{2} \frac{2}{x+1} \, dx \) gives \( 2\ln\frac{3}{2} \),
• \( \int_{1}^{2} \frac{3}{x+2} \, dx \) calculates to \( 3\ln\frac{4}{3} \).

These results combine to reflect the overall area calculations within the prescribed limits.

Logarithmic integration is an elegant method used to solve integrals of the form \( \int \frac{1}{u} \, du \), which results in a natural logarithm of the absolute value of \(u\).### Basics of Logarithmic IntegrationWhen we encounter an expression like \( \frac{1}{x} \), its integral naturally transforms via:\[\int \frac{1}{x} \, dx = \ln|x| + C.\]This is the foundational formula for integrating functions that resolve into logarithmic forms.### Application to ExerciseEach partial fraction in our example follows this principle:

• \( \int \frac{1}{x} \, dx = \ln|x|, \)
• \( \int \frac{2}{x+1} \, dx = 2 \ln|x+1|, \)
• \( \int \frac{3}{x+2} \, dx = 3 \ln|x+2|. \)

When we integrate these fractions within their definite bounds, they yield logarithmic expressions, showcasing how logarithmic integration simplifies complex integrands.