Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1–40, use the method of partial fraction decomposition to perform the required integration. $$ \int \frac{3 x+2}{x^{3}+3 x^{2}+3 x+1} d x $$

Step-by-Step Solution

Answer

Integrate: \( -\frac{3}{x+1} + \frac{1}{2(x+1)^2} + C \).

Step 1: Factor the Denominator

First, identify the factors of the cubic polynomial \( x^3 + 3x^2 + 3x + 1 \). By attempting polynomial division or using synthetic division, you can determine that \( x^3 + 3x^2 + 3x + 1 = (x+1)(x^2 + 2x + 1) = (x+1)^3 \). This means the cubic polynomial simplifies to a repeated linear factor.

Step 2: Set Up Partial Fractions

Since the denominator is \( (x+1)^3 \), the partial fraction decomposition takes the form \( \frac{3x+2}{(x+1)^3} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)^3} \). This will allow us to express the fraction as a sum of fractions with simpler denominators.

Step 3: Clear the Denominators

Multiply through by \( (x+1)^3 \) to eliminate the denominators from the equation, giving you: \( 3x + 2 = A(x+1)^2 + B(x+1) + C \). This simplifies to a polynomial equation matching both sides.

Step 4: Expand and Simplify

Expand the right side: \( A(x^2 + 2x + 1) + B(x+1) + C = Ax^2 + 2Ax + A + Bx + B + C \). This results in \( Ax^2 + (2A+B)x + (A+B+C) \). Collect terms according to powers of \( x \).

Step 5: Solve for Constants A, B, and C

Compare the coefficients from both sides: 1. \( A = 0 \) (No \( x^2 \) term on the left)2. \( 2A + B = 3 \) (Coefficient of \( x \))3. \( A + B + C = 2 \) (Constant term)From \( A = 0 \), substitute into other equations to solve for \( B \) and \( C \):\( B = 3 \), \( 0 + 3 + C = 2 \). Thus, \( C = -1 \).

Step 6: Rewrite the Integral

Substitute \( A \), \( B \), and \( C \) back into the partial fractions: \[ \int \left( \frac{3}{(x+1)^2} - \frac{1}{(x+1)^3} \right) dx \]. Separate the integrals: \( \int \frac{3}{(x+1)^2} dx - \int \frac{1}{(x+1)^3} dx \).

Step 7: Integrate Each Term

For \( \int \frac{3}{(x+1)^2} dx \), the antiderivative is \( -\frac{3}{x+1} \). For \( \int \frac{1}{(x+1)^3} dx \), the antiderivative is \( \frac{1}{2(x+1)^2} \). Thus, the integral is \(-\frac{3}{x+1} + \frac{1}{2(x+1)^2} + C \).

Key Concepts

Integration Techniques

When faced with complex integrals, such as \[ \int \frac{3x+2}{x^3+3x^2+3x+1} dx, \]one effective approach is the method of partial fraction decomposition. This technique simplifies the process by breaking down the integrand into simpler fractions that are easier to integrate.
This method becomes essential when dealing with rational functions, where the integral features a polynomial numerator over a polynomial denominator.
Using partial fraction decomposition:

• First, factor the polynomial in the denominator.
• Set up the partial fraction, accounting for the type and multiplicity of factors.
• Solve for constants that simplify the integral into manageable parts with known antiderivatives.

The strategic application of these steps makes integration much more approachable.

Factorization of Polynomials

Factorizing the polynomial in the denominator is the first step in partial fraction decomposition.
In our exercise, the polynomial is \( x^3 + 3x^2 + 3x + 1 \).
Through trial methods such as synthetic division, we establish its factorization as \((x+1)^3\).
Understanding polynomial factorization involves:

• Recognizing patterns or using division methods to split the polynomial into linear or quadratic factors.
• Handling repeated roots, which result in multiple factors such as \((x+1)^3\).

Factorization simplifies the decomposition process by verifying the factors we need for partial fraction setup.

Antiderivative Calculation

After setting up the partial fraction decomposition and solving for constants, you substitute these into the integral.
For example, converting \[\int \frac{3x+2}{(x+1)^3} dx\] to \[\int \left(\frac{3}{(x+1)^2} - \frac{1}{(x+1)^3}\right) dx.\]The next step is calculating the antiderivatives of these simpler fractions separately.Here:

• The antiderivative of \( \frac{3}{(x+1)^2} \) is \( -\frac{3}{x+1} \).
• The antiderivative of \( \frac{1}{(x+1)^3} \) is \( \frac{1}{2(x+1)^2} \).

Combining these results using integration rules yields the complete solution, which also includes adding a constant of integration \( C \).

Rational Functions

In calculus, rational functions are quotients of two polynomials, such as \[\frac{3x+2}{x^3+3x^2+3x+1}.\]Analyzing rational functions through the lens of calculus involves operations such as differentiation and integration, both of which may require managing the interaction of numerator and denominator polynomials.
The goal is typically simplification, facilitating integration or further analysis.
In this context, to integrate rational functions:

• First, ensure the polynomial degree in the numerator is less than the denominator. If not, perform polynomial long division.
• Apply partial fraction decomposition as a strategy to break the rational function down into simpler fractions.

This makes it significantly easier to find antiderivatives, making rational functions manageable within integral calculus.