Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{3}-8 x^{2}-1}{(x+3)\left(x^{2}-4 x+5\right)} d x\)

Short answer

The integral is \( 2 \ln|x+3| - \frac{1}{2} \ln|x^2-4x+5| + C \).

Step-by-step solution

Understanding the Problem

We are asked to integrate the function \( \frac{x^{3}-8x^{2}-1}{(x+3)(x^{2}-4x+5)} \) using partial fraction decomposition. This involves expressing the integrand as a sum of simpler fractions that can be easily integrated.

Setup the Partial Fraction Decomposition

The expression \( \frac{x^{3}-8x^{2}-1}{(x+3)(x^{2}-4x+5)} \) can be decomposed into partial fractions. We write it as \( \frac{A}{x+3} + \frac{Bx+C}{x^2-4x+5} \). The next step is to determine coefficients \( A, B, \) and \( C \).

Form the Equation by Equating Numerators

Multiply both sides by the common denominator \((x+3)(x^{2}-4x+5)\) to get \( x^{3}-8x^{2}-1 = A(x^{2}-4x+5) + (Bx+C)(x+3) \). We will solve this equation to find the values of \(A\), \(B\), and \(C\).

Expand and Collect Like Terms

Expand the right-hand side: \[ A(x^{2}-4x+5) = Ax^2 - 4Ax + 5A \] and \[ (Bx + C)(x + 3) = Bx^2 + 3Bx + Cx + 3C \]. Collect like terms: \[ x^{3} - 8x^{2} - 1 = (A + B)x^2 + (-4A + 3B + C)x + (5A + 3C) \].

Solve the System of Equations

Equate coefficients from the right-hand side with \( x^3 - 8x^2 - 1 \): 1. \( A + B = 1 \) from \( x^2 \)2. \(-4A + 3B + C = -8 \) from \( x \)3. \( 5A + 3C = -1 \) from the constant term. Solve this system of equations for \( A, B, \) and \( C \).

Calculation of Coefficients

From equation \(1\): \( B = 1 - A \).Plug this into equations \(2\) and \(3\):\(-4A + 3(1-A) + C = -8\)\(5A + 3C = -1\) Solve these linear equations to find \( A = 2, B = -1, C = -3 \).

Write the Partial Decomposition

Using \(A, B, \) and \(C\), the decomposition is:\( \frac{x^{3}-8x^{2}-1}{(x+3)(x^{2}-4x+5)} = \frac{2}{x+3} - \frac{x+3}{x^2-4x+5} \)

Integrate Each Term Separately

We now integrate each term separately:- For \( \int \frac{2}{x+3} dx \), the integral is \( 2 \ln|x+3| + C_1 \). - For \( \int \frac{-x-3}{x^2-4x+5} dx \), use the substitution \(u = x^2-4x+5\), then integrate to find \( \ln|u| \).

Complete the Integration

Using \( u = x^2 - 4x + 5 \) in substitution for the second term, solve the integrals and simplify the result:\( \int \frac{-x-3}{x^2-4x+5} \approx -\frac{1}{2} \ln|x^2-4x+5| + C_2 \). Your final integrated result is:\[ 2 \ln|x+3| - \frac{1}{2} \ln|x^2-4x+5| + C \] where \( C = C_1 + C_2 \).

Key concepts

Integration Techniques

Integration is a way to find the area under a curve. It's essential in calculus and helps solve problems involving continuous sums. One technique is **partial fraction decomposition**, which simplifies complex rational functions. When you see a function like \( \int \frac{x^{3}-8 x^{2}-1}{(x+3)(x^{2}-4 x+5)} \, dx \), integrating directly can be daunting. Instead, we simplify it by breaking it into easier parts using partial fractions.

This method involves expressing the complex fraction as a sum of simpler fractions. These simpler fractions are easier to integrate. The core idea is to transform the integration into a sum of terms that involve basic integrals, which are more manageable to solve. The partial fractions turn the problem into smaller, simpler problems. You then integrate each one separately, making the task straightforward.

Partial fraction decomposition is especially useful for integrating rational functions, where the degree of the numerator is less than the degree of the denominator. By using this method effectively, you can break down intricate integrations into basic components, each of which is much easier to handle.

Algebraic Manipulation

Algebraic manipulation is a helpful tool for solving complex problems. It involves rearranging and simplifying equations to make them more accessible. In partial fraction decomposition, algebraic manipulation plays a crucial role in setting up the problem correctly.

For example, when we break down \( \frac{x^{3}-8x^{2}-1}{(x+3)(x^{2}-4x+5)} \), we use algebraic manipulation to write it as a sum of partial fractions:\( \frac{A}{x+3} + \frac{Bx+C}{x^2-4x+5} \).

This step involves creatively manipulating the form of the function and involves:

• Expanding terms: Distributing and simplifying expressions to align powers of variables on both sides.
• Collecting like terms: Grouping similar terms to make it easier to see relationships between coefficients.

This process helps us form a system of equations that we can solve to find the constants needed for the partial fraction decomposition. This intricate dance of adjusting, simplifying, and rearranging terms ensures that the integration process becomes gar simpler once algebraic manipulation is correctly executed.

Systems of Equations

A system of equations is a collection of equations with multiple variables. Solving them helps find the values of these variables that satisfy all equations simultaneously.

In partial fraction decomposition, after rewriting the fraction, we end up with a system of equations from matching coefficients. For the decomposition \( \frac{x^{3}-8x^{2}-1}{(x+3)(x^{2}-4x+5)} \), we derived the following system:

• Equation 1: \( A + B = 1 \)
• Equation 2: \(-4A + 3B + C = -8 \)
• Equation 3: \(5A + 3C = -1 \)

Solving this system tells us the values of \(A, B,\) and \(C\).

We use standard techniques like substitution and elimination. Solving the system of equations provides the necessary coefficients to express the original integral as separate, simpler integrals. This process underscores the importance of analytical and problem-solving skills, enabling us to break down complex problems into achievable pieces efficiently.