Question

Use the method of partial fractions to evaluate each of the following integrals. \(\int \frac{d x}{x^{4}-10 x^{2}+9}\)

Short answer

The integral evaluates to multiple logarithmic terms, representing each partial fraction component.

Step-by-step solution

Factor the Denominator

To use partial fractions, first factor the denominator of the integrand. Set \( x^4 - 10x^2 + 9 = 0 \) and let \( y = x^2 \), then rewrite as \( y^2 - 10y + 9 = 0 \). This is a quadratic equation which can be factored into \( (y - 9)(y - 1) = 0 \). Substituting back \( y = x^2 \), we have \( (x^2 - 9)(x^2 - 1) \), which further factors as \( (x - 3)(x + 3)(x - 1)(x + 1) \).

Set Up Partial Fractions

Express \( \frac{1}{(x - 3)(x + 3)(x - 1)(x + 1)} \) as a sum of partial fractions: \[ \frac{A}{x - 3} + \frac{B}{x + 3} + \frac{C}{x - 1} + \frac{D}{x + 1} \].

Clear the Denominator

Multiply through by the common denominator \( (x - 3)(x + 3)(x - 1)(x + 1) \) to obtain: \[ 1 = A(x + 3)(x - 1)(x + 1) + B(x - 3)(x - 1)(x + 1) + C(x - 3)(x + 3)(x + 1) + D(x - 3)(x + 3)(x - 1) \].

Solve for Coefficients

Find the coefficients \( A, B, C, \) and \( D \) by substituting convenient values for \( x \) that simplify the equation. For example, set \( x = 3 \) to find \( A \), \( x = -3 \) to find \( B \), \( x = 1 \) to find \( C \), and \( x = -1 \) to find \( D \).

Integrate Each Term

Once the coefficients are known, integrate each term separately: \[ \int \frac{A}{x-3} \, dx, \int \frac{B}{x+3} \, dx, \int \frac{C}{x-1} \, dx, \int \frac{D}{x+1} \, dx \]. Use the formula \( \int \frac{1}{x-a} \, dx = \ln |x-a| + C \).

Combine and Simplify

Combine the results of the integrals obtained in Step 5 to write the final solution to the integral. Make sure to include the constant of integration \( C \) in the final step.

Key concepts

Integration Techniques

When faced with the task of evaluating an integral, especially one involving a rational function like \( \int \frac{d x}{x^{4}-10 x^{2}+9} \), you need to select the integration technique that fits best. Partial fraction decomposition is a popular method here. It involves breaking down a complex rational expression into simpler fractions that are easier to integrate.

• Identify the type of expression in the integral.
• Choose an appropriate method, such as substitution, partial fractions, or integration by parts.
• Use partial fractions for rational expressions, especially when the degree of the polynomial in the numerator is less than the degree in the denominator.

In our exercise, partial fractions help turn a daunting polynomial into manageable parts. By analyzing the integral and applying the right techniques, you can transform complex problems into straightforward tasks.

Factoring Polynomials

Factoring polynomials is often the first step in dealing with integrals involving rational expressions. Let's break down what factoring means and how it's applied.
Factoring involves rewriting a polynomial as a product of simpler polynomials. It simplifies expressions and reveals root information, vital for partial fraction decomposition.

Steps to Factor

• Recognize and apply polynomial identities, e.g., quadratic, cubic formulas, or difference of squares.
• Rewrite the polynomial, focusing on possible simplifications by division or grouping.
• Check your factorization by multiplying back to ensure accuracy.

In the exercise, the polynomial \( x^4 - 10x^2 + 9 \) is factored into simpler terms using substitution and difference of squares: first as \( x^2 - 9 \), then as \((x - 3)(x + 3)\), and \( x^2 - 1 \) as \((x - 1)(x + 1)\). This reveals the structure needed for decomposition.

Rational Functions

Rational functions are key players in calculus, particularly in integration tasks. A rational function can be expressed as the quotient of two polynomials.
In essence, the function form \( \frac{P(x)}{Q(x)} \) shows a numerator \( P(x) \) and a denominator \( Q(x) \), where both are polynomials.

Features of Rational Functions

• Asymptotes: Where the graph approaches a line but never touches, can be vertical or horizontal.
• Intercepts: Points where the graph crosses the axes. Find these by setting \( x \) and \( y \) to zero appropriately.
• Domain: All real numbers except those that make \( Q(x) \)=0.

For integrals involving rational functions, partial fraction decomposition helps separate pieces based on polynomial factorization. This way, evaluating the integral becomes simpler and more systematic.

Integrals of Rational Expressions

Integrating rational expressions is a fundamental skill in calculus. When the expression involves polynomials, partial fraction decomposition becomes an invaluable tool.
The integration process naturally flows once a rational expression is decomposed into simpler fractions.

Steps for Integration

• Decompose the rational expression using partial fractions.
• Integrate each fraction separately using basic logarithmic identities such as \( \int \frac{1}{x-a} \, dx = \ln |x-a| + C \).
• Combine the resulting integrals and include the constant of integration \( C \) at the end.

In our exercise, after factoring and decomposition, each fraction is integrated separately. By integrating using the natural log formula, you obtain results free of the original complex polynomial structure. This reflects the power of partial fractions in simplifying integration challenges.