Problem 24 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{x^{6}}{(x-2)^{2}(1-x)^{5}} d x $$

Step-by-Step Solution

Answer

Decompose the fraction, solve for constants, and integrate each term separately.

Step 1: Rewrite the Denominator

To perform partial fraction decomposition, start by considering the denominator. The denominator is \((x-2)^2(1-x)^5\). Notice that we can rewrite \((1-x)\) as \((-1)(x-1)\), hence \((1-x)^5 = (-1)^5(x-1)^5 = -(x-1)^5\). So the denominator becomes \((x-2)^2(x-1)^5\).

Step 2: Set Up the Partial Fraction Decomposition

Decompose the rational function into partial fractions. Assume a form for the decomposition as follows:\[\frac{x^6}{(x-2)^2(x-1)^5} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x-1} + \frac{D}{(x-1)^2} + \frac{E}{(x-1)^3} + \frac{F}{(x-1)^4} + \frac{G}{(x-1)^5}\] This form considers linear terms for \((x-2)\) and up to fifth-degree terms for \((x-1)\).

Step 3: Clear the Denominator

Multiply both sides by the common denominator \((x-2)^2(x-1)^5\) to clear the denominators:\[x^6 = A(x-2)(x-1)^5 + B(x-1)^5 + C(x-2)^2(x-1)^4 + D(x-2)^2(x-1)^3 + E(x-2)^2(x-1)^2 + F(x-2)^2(x-1) + G(x-2)^2\]

Step 4: Expand and Combine Like Terms

Expand the right-hand side and collect like terms to solve for constants A, B, C, D, E, F, and G. This involves significant algebraic manipulation, where you match coefficients of corresponding powers of \(x\) on both sides.

Step 5: Solve for Constants

Solve the system of equations formed by equating the coefficients from both sides of the equation from the expanded form. Substitute specific values for \(x\) that simplify the equation, such as \(x=2\) and \(x=1\), to ease calculation for particular constants. This is generally done using substitution, comparison, or by setting up equations for each term.

Step 6: Integrate Each Term

Once you've found the constants, rewrite the original integral with these partial fractions:\[\int \left(\frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{C}{x-1} + \ldots + \frac{G}{(x-1)^5} \right) dx\]Each term can be integrated using simple integration rules:- \( \int \frac{1}{x-a} dx = \ln|x-a| + C \)- \( \int \frac{1}{(x-a)^n} dx = \frac{-(x-a)^{1-n}}{1-n} + C \) for \(n > 1\)Evaluate these integral terms individually.

Key Concepts

Integration Techniques

Integration is a fundamental tool in calculus used to calculate areas under curves and other quantities. One powerful technique in dealing with complex rational functions involves partial fraction decomposition. When faced with integrating a function like \( \int \frac{x^6}{(x-2)^2(1-x)^5} \ dx \), direct integration may not be possible.
Decomposing it into simpler fractions allows each term to be integrated individually with known integration formulas. Using partial fraction decomposition, you'll express a complex rational function as a sum of simpler fractions like \( \frac{A}{x-2} \) and \( \frac{B}{(x-1)^5} \), making the integration much more feasible.
These techniques simplify the calculation and allow for the application of basic integration rules such as:

• \( \int \frac{1}{x-a} \ dx = \ln|x-a| + C \)
• \( \int \frac{1}{(x-a)^n} \ dx = \frac{-(x-a)^{1-n}}{1-n} + C \), for \( n > 1 \)

By mastering these methods, the integration of rational functions becomes straightforward, opening up a wide array of problem-solving opportunities.

Algebraic Manipulation

In algebra, manipulation involves the systematic rearrangement of expressions to simplify problems or equations. For partial fraction decomposition, it begins with rewriting the denominator.
Given a denominator in a rational function like \( (x-2)^2(1-x)^5 \), note that \( 1-x \) can be rewritten as \( -(x-1) \). Hence, the expression \( (1-x)^5 \) transforms into \( -(x-1)^5 \), and consequently, the full denominator becomes \( (x-2)^2(x-1)^5 \).
When finding the coefficients for the partial fractions, involve equations derived from clearing the denominator. This step forms a system of equations based on powers of \( x \). Equating coefficients of both sides requires:

• Expanding product terms to reveal individual coefficients
• Solving the resulting linear equations systematically for unknown constants

This methodical manipulation provides the foundation for decomposing complex functions, which eases further integration processes.

Rational Functions

Rational functions are a form of expression where one polynomial divides another. They are written as \( \frac{P(x)}{Q(x)} \), with the degree of \( P(x) \) and \( Q(x) \) determining the function's behavior.
In partial fraction decomposition, we decompose a complex rational function like \( \frac{x^6}{(x-2)^2(1-x)^5} \) into simpler terms such as \( \frac{A}{x-2} \) or \( \frac{B}{(x-1)^5} \). This process is essential for integrating rational functions, especially when \( Q(x) \) is a product of linear or irreducible quadratic components.
Understanding rational functions allows mathematicians to:

• Predict discontinuities or asymptotic behavior
• Determine roots and factors affecting integration and calculus operations

By simplifying rational functions into manageable parts, partial fraction decomposition facilitates the further steps needed in complex integral calculus.

Mathematical Problem Solving

Mathematical problem solving is not just about finding the answer but also about understanding the process. In the context of partial fraction decomposition, the solution begins by decomposing a challenging integral into simpler, more manageable fractions.
In tackling problems like \( \int \frac{x^6}{(x-2)^2(1-x)^5} \ dx \), follow a systematic approach:

• Identify the type of rational function
• Break it into simpler parts through partial fraction decomposition
• Find unknown coefficients by setting up and solving algebraic equations
• Integrate each resulting fraction independently

Each step demands careful calculation and attention to algebraic details. By deeply understanding each stage, students can tackle increasingly complex problems with confidence and clarity.