Problem 46 - Calculus for Biology and Medicine

Question

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{2 x^{3}-x-1}{x^{2}(x+1)^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln |x| - \frac{1}{x} + \ln |x+1| + C \).

Step 1: Identify the Denominator Polynomial

The first step in partial fraction decomposition is to identify the polynomial in the denominator. The given expression is \( \frac{2x^3 - x - 1}{x^2(x+1)^2} \). The denominator \( x^2(x+1)^2 \) tells us that we will have two types of terms in the decomposition: terms with \( x \) and terms with \( x+1 \).

Step 2: Set Up the Partial Fraction Decomposition

We decompose the integrand into partial fractions. Based on the factors in the denominator, we write:\[ \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} + \frac{D}{(x+1)^2} \]These correspond to the factors \( x^2 \) and \( (x+1)^2 \) in the original denominator.

Step 3: Clear the Denominator and Combine Terms

Multiply through by the common denominator \( x^2(x+1)^2 \) to clear the fractions:\[ 2x^3 - x - 1 = A x(x+1)^2 + B (x+1)^2 + C x^2(x+1) + D x^2 \]

Step 4: Expand and Collect Like Terms

Expand each term on the right-hand side of the equation:- \( A(x)(x^2 + 2x + 1) = Ax^3 + 2Ax^2 + Ax \)- \( B(x^2 + 2x + 1) = Bx^2 + 2Bx + B \)- \( Cx^2(x + 1) = Cx^3 + Cx^2 \)- \( Dx^2 = Dx^2 \)Combine like terms, resulting in:\[ (A + C)x^3 + (2A + B + C + D)x^2 + (A + 2B)x + B \]

Step 5: Solve for Unknown Coefficients

Equate the coefficients from the expanded polynomial and the original numerator:- For \( x^3 \): \( A + C = 2 \)- For \( x^2 \): \( 2A + B + C + D = 0 \)- For \( x \): \( A + 2B = -1 \)- Constant term: \( B = -1 \)Solve these equations to find: \( A = 1, B = -1, C = 1, D = 0 \).

Step 6: Rewrite the Integral with Decomposed Fractions

Use the solved coefficients to express the original integral as:\[ \int \left( \frac{1}{x} - \frac{1}{x^2} + \frac{1}{x+1} \right) \, dx \]

Step 7: Integrate Each Term Separately

Now integrate each term separately:1. \( \int \frac{1}{x} \, dx = \ln |x| + C_1 \)2. \( \int -\frac{1}{x^2} \, dx = \frac{1}{x} + C_2 \)3. \( \int \frac{1}{x+1} \, dx = \ln |x+1| + C_3 \)Combine the results to get the overall integral.

Step 8: Write the Final Solution

Combine all the integrated terms:\[ \ln |x| - \frac{1}{x} + \ln |x+1| + C \]

Key Concepts

Integration Techniques

Integration techniques are essential methods in calculus for computing the area under curves, which is represented by integrals. The problem given here uses partial fraction decomposition to integrate a complex rational function. This technique is vital when dealing with rational functions, which are ratios of polynomials. In such cases, directly integrating the function might be challenging due to its complexity.

To simplify the process, partial fraction decomposition breaks down the complex rational function into a sum of simpler fractions that can be more easily integrated individually. Once the expression is decomposed, each fraction can usually be integrated using basic formulas. For instance, fractions like \( \frac{1}{x} \) can be integrated using the natural logarithm, while terms like \( \frac{1}{x^2} \) require polynomial integration rules.

This exercise exemplifies the use of partial fraction decomposition in easing the integration process by transforming a challenging problem into a series of straightforward steps, illustrating the ingenuity of mathematical techniques in problem solving.

Mathematical Problem Solving

Mathematical problem solving involves a series of logical and analytical skills to find solutions for various mathematical questions. In our given partial fraction decomposition problem, we follow a stepwise approach to reach the solution. The process begins with identifying and analyzing the polynomial given in the denominator. Understanding the structure of the given problem is crucial for correctly setting up the partial fractions.

In this exercise, the denominator is \( x^2(x+1)^2 \), which hints at how we will split the overall fraction. We decompose the function into partial fractions using hypothetical constants, which are placeholders that we will solve for later. This approach is systematic and logical:

• Make an educated guess or hypothesis based on the structure.
• Solve for unknowns by equating coefficients between expanded forms.
• Use logical mathematical steps to arrive at the correct decomposition.

Once the partial fractions are derived, integrating each term is a straightforward task, demonstrating how breaking down complex problems into simpler ones can aid in efficient problem solving.

Polynomial Long Division

Polynomial long division is similar to numerical long division but involves dividing polynomials. It's a technique used in some cases of rational functions, especially when the numerator's degree is greater or equal to the denominator's degree.

In the context of partial fraction decomposition, polynomial long division helps in simplifying the function before decomposition if necessary. However, for this particular exercise, long division wasn't required because the numerator didn't exceed the denominator's polynomial degree.

When applicable, polynomial long division will divide the polynomials completely or provide a remainder, resulting in a simpler expression. Understanding this skill is critical in situations where simplification is needed before decomposition or integration. It is an important tool in the arsenal of mathematical techniques available to solve complex rational functions.