Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\(-2 \ln |x| + 2 \ln |x^2 + 4| + \frac{1}{2}\tan^{-1}(x/2) + C\).

Step 1: Factor the Denominator

First, identify and factor the denominator. The denominator is \(x^3 + 4x\). This can be factored as \(x(x^2 + 4)\).

Step 2: Set Up Partial Fractions

Write the expression \(\frac{2x^2 + x - 8}{x(x^2 + 4)}\) as a sum of partial fractions: \(\frac{A}{x} + \frac{Bx + C}{x^2 + 4}\).

Step 3: Find Coefficients

Multiply both sides by the denominator \(x(x^2+4)\) to eliminate fractions: \(2x^2 + x - 8 = A(x^2 + 4) + (Bx + C)x\). Expand and collect like terms: \(2x^2 + x - 8 = (A + B)x^2 + Cx + 4A\). Equate coefficients: \(A + B = 2\), \(C = 1\), and \(4A = -8\).

Step 4: Solve for Coefficients

Solve the equations: From \(4A = -8\), we get \(A = -2\). Then from \(A + B = 2\), substitute \(A = -2\) to find \(B = 4\). We already have \(C = 1\).

Step 5: Rewrite the Integral with Partial Fractions

Substitute back into partial fractions: \(\int \left(\frac{-2}{x} + \frac{4x+1}{x^2 + 4}\right) dx\).

Step 6: Integrate Each Term Separately

Integrate term by term: \(\int \frac{-2}{x} dx = -2 \ln |x| + C_1\) and \(\int \frac{4x}{x^2 + 4} dx + \int \frac{1}{x^2 + 4} dx\). This is \(2 \ln |x^2 + 4| + C_2 + \frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right) + C_3\).

Step 7: Combine the Results

The combined integral is \(-2 \ln |x| + 2 \ln |x^2 + 4| + \frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right) + C\), where \(C = C_1 + C_2 + C_3\).

Key Concepts

Integration Techniques

In calculus, integrating rational functions can be challenging, especially when dealing with complex expressions. Partial fraction decomposition is one technique that simplifies this process. This method involves breaking down a complicated rational function into simpler fractions that are easier to integrate.

The general idea is to express the integrand as a sum of simpler fractions. For instance, in the integral \( \int \frac{2x^2 + x - 8}{x^3 + 4x} \, dx \), after factoring the denominator and setting up fractions, we use algebra to solve for coefficients. This allows us to integrate each term separately.

Here’s what makes this approach beneficial:

• Simplifies rational functions into easily manageable parts.
• Facilitates integration of complex functions by dealing with simpler expressions.
• Transforms a difficult integral into smaller tasks.

Partial fraction decomposition is crucial in calculus as an integration technique, helping tackle a diverse range of integration problems.

Calculus Integrals

Integrating a function involves finding a function, called the antiderivative, whose derivative gives back the original function. This process works like reversing differentiation, allowing us to determine the area under curves in calculus.

Calculus integrals appear in two main forms: definite and indefinite. Indefinite integrals, like the one here, \( \int \frac{2x^2 + x - 8}{x^3 + 4x} \, dx \), do not have specified bounds and include a constant of integration \( C \). Definite integrals, by contrast, compute the area between the graph of the function and the interval from \( a \) to \( b \). These TWO forms of integration are pivotal in understanding the behavior of functions.

In practice:

• Indefinite integrals help in finding general solutions to differential equations.
• Definite integrals are used to compute exact areas, volumes, and in various applications across physics and engineering.

Grasping the fundamentals of calculus integrals is essential for advancing in mathematics and tackling complex problems in applied fields.

Rational Functions Integration

Rational functions are quotients of polynomial functions. Integrating these functions often involves techniques that simplify their structure. Partial fraction decomposition is highly effective here, particularly when the degree of the numerator is less than that of the denominator.

In our example, \( \int \frac{2x^2 + x - 8}{x^3 + 4x} \, dx \), the degree of the numerator, 2, is less than the denominator, 3. This characteristic is crucial, as it allows us to employ partial fraction decomposition efficiently. The decomposition results in simpler fractions that can then be integrated separately.

Key points to consider:

• Factor the denominator to identify which partial fractions to construct.
• Set up an equation using coefficients for each term of the partial fractions.
• Solve for each coefficient to complete the decomposition.

This approach not only eases the integration process but also provides a structured method for working with complex rational functions.