Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{2 x^{3}+5 x^{2}+16 x}{x^{5}+8 x^{3}+16 x} d x\)

Short answer

Factor the denominator, set up partial fractions, solve for coefficients, and integrate each term.

Step-by-step solution

Factor the Denominator

The first step is to factor the denominator of the integrand: \(x^5 + 8x^3 + 16x\). Notice that each term in the polynomial has an \(x\), so we can factor out \(x\) leaving us with \(x(x^4 + 8x^2 + 16)\). Further factor \(x^4 + 8x^2 + 16 = (x^2 + 4)^2\). Thus, the factored form of the denominator is \(x(x^2 + 4)^2\).

Set Up Partial Fractions

Now, set up the partial fraction decomposition for the expression \(\frac{2x^3 + 5x^2 + 16x}{x(x^2 + 4)^2}\). We assume a decomposition of the form: \[ \frac{A}{x} + \frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2} \].

Clear the Dominator

Multiply both sides by the denominator \(x(x^2 + 4)^2\) to clear fractions and obtain: \[ 2x^3 + 5x^2 + 16x = A(x^2 + 4)^2 + (Bx + C)x(x^2 + 4) + (Dx + E)x \].

Expand and Collect Like Terms

Expand the right-hand side of the equation to collect like terms. After expansion, compare coefficients of the resulting polynomials on both sides to solve for the unknowns \(A, B, C, D, E\).

Solve for Coefficients

Equate the coefficients of powers of \(x\) on both sides to form a system of equations, then solve for \(A, B, C, D, E\). For instance: Equating \(x^3\) terms yields one equation, \(x^2\) terms yields another, and so forth.

Integrate Each Term

Once values for \(A, B, C, D, E\) are found, the integral can be written as a sum of simpler fractions: \(\int \frac{A}{x} dx + \int \frac{Bx + C}{x^2 + 4} dx + \int \frac{Dx + E}{(x^2 + 4)^2} dx\). Integrate each term separately, potentially using substitution techniques or recognizing standard integral forms.

Key concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used in calculus to break down a complex rational expression into simpler fractions that are easier to integrate. This method is especially helpful when dealing with rational functions, which are ratios of polynomials.

When performing partial fraction decomposition, start by ensuring the degree of the numerator is less than the degree of the denominator. If not, perform polynomial division first. Then, factor the denominator completely. In this exercise, the denominator, \(x^5 + 8x^3 + 16x\), is factored into \(x(x^2 + 4)^2\).

• Write the original expression as a sum of fractions, with unknown coefficients. For example, \(\frac{A}{x} + \frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2}\).
• Clear the fractions by multiplying each term by the common denominator.
• Expand and collect like terms on both sides of the equation.
• Solve for the unknown coefficients by comparing coefficients of each power of \(x\).

This decomposition makes integrating the expression manageable by breaking it into simpler parts.

Integral Calculus

Integral calculus focuses on the concept of the integral, which aims to find the accumulation or total quantity when continuous change is considered. It is often seen as the reverse operation of differentiation.

In the context of this exercise, after decomposing the rational function into partial fractions, each fraction is integrated separately. This involves:

• Recognizing standard forms of integrals, such as \(\frac{1}{x}\), the integral of which is a natural logarithm \(\ln |x|\).
• Using substitution or other integration techniques for more complex terms.
• Summing the results of all separate integrals to find the total integral of the original rational function.

Integral calculus provides powerful tools for finding areas, volumes, and other quantities that involve accumulation, which are essential across physics, engineering, and other fields.

Factoring Polynomials

Factoring polynomials is a critical step when working with rational functions, particularly in partial fraction decomposition. The goal is to express a polynomial as a product of simpler, irreducible polynomials.

In this exercise, we start by factoring the polynomial \(x^5 + 8x^3 + 16x\). Factoring out the greatest common factor (GCF) first, in this case, \(x\), simplifies the polynomial to \(x(x^4 + 8x^2 + 16)\).

• The expression \(x^4 + 8x^2 + 16\) can be recognized as a perfect square trinomial and is factored further into \((x^2 + 4)^2\).
• This kind of factorization is crucial in breaking down the rational expression for partial fraction decomposition.

These techniques simplify the process of integration by making the algebra manageable, setting the stage for clear, coherent decomposition.

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. They are found extensively in calculus, algebra, and other areas of mathematics due to their application in modeling real-world systems.

Understanding rational functions' properties is crucial for solving integrals, particularly when involving polynomials in both the numerator and the denominator.

• Before integrating rational functions, ensure they are simplified as much as possible, through cancellation or other algebraic techniques.
• Use partial fraction decomposition to transform the complexity of these functions into simpler fractions.
• Such functions can be processed in integrals to analyze behavior between finite intervals or to understand asymptotic behavior as variables approach infinity.

Profound understanding of rational functions empowers problem-solving in various mathematical disciplines and enhances one's ability to tackle problems involving continuous change.