Problem 52 - Calculus Early Transcendentals

Question

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int \frac{d x}{x\left(a^{2}-x^{2}\right)^{2}}$$

Step-by-Step Solution

Answer

Based on the given step by step solution, create a short answer for the problem. To evaluate the indefinite integral of the function $$\frac{1}{x(a^2 - x^2)^2}$$ with respect to x, we first applied partial fraction decomposition and obtained the coefficients A, B, and C. After rewriting the integrand as a sum of simpler fractions, we integrated each fraction to get the final result: $$\int \frac{d x}{x\left(a^{2}-x^{2}\right)^{2}} = \frac{1}{a^4}\left[\ln |x| - \frac{1}{2a} \ln\left|\frac{a + x}{a - x}\right| + \frac{x}{a^2(a^2 - x^2)} \right] + C$$ where C is the constant of integration.

Step 1: Recognize the integral form

The given integral has the following form: $$\int \frac{d x}{x\left(a^{2}-x^{2}\right)^{2}}$$ This integral has a fraction with a quadratic polynomial in the denominator, raised to the power of 2.

Step 2: Apply partial fraction decomposition

We need to rewrite the integrand as the sum of simpler fractions. The partial fraction decomposition for this integrand would be of the form: $$\frac{1}{x(a^2 - x^2)^2} = \frac{A}{x} + \frac{B}{(a^2 - x^2)} + \frac{C}{(a^2 - x^2)^2}$$ To solve for A, B, and C, we first need to clear the denominators by multiplying both sides by $$x(a^2 - x^2)^2$$: $$1 = A(a^2 - x^2)^2 + Bx(a^2 - x^2) + Cx$$

Step 3: Solving for the coefficients A, B, and C

We need to find the values of A, B, and C that satisfy this equation. We can do this by setting x = 0 and plugging in points to create a system of linear equations. At x = 0, we get: $$1 = A(a^2)^2$$ $$ A = \frac{1}{a^4}$$ Now we need to find the values for B and C. We can use x = a and x = -a to get two more equations: For x = a: $$-1 = B(a^4)$$ $$ B = -\frac{1}{a^4}$$ For x = -a: $$1 = C(-a)^4$$ $$ C = \frac{1}{a^4}$$ Now that we have the coefficients, we can rewrite the integrand as: $$\frac{1}{x\left(a^{2}-x^{2}\right)^{2}} = \frac{\frac{1}{a^4}}{x} - \frac{\frac{1}{a^4}} {(a^2 - x^2)} + \frac{\frac{1}{a^4}} {(a^2 - x^2)^2}$$

Step 4: Integrate each of the fractions

Now we need to integrate each of these simple fractions with respect to x: $$\int \frac{d x}{x\left(a^{2}-x^{2}\right)^{2}} = \int \frac{1}{a^4x} dx - \int \frac{1}{a^4(a^2 - x^2)}dx + \int \frac{1}{a^4(a^2 - x^2)^2}dx$$ $$ = \frac{1}{a^4}\left[\int \frac{1}{x} dx - \int \frac{1}{a^2 - x^2}dx + \int \frac{1}{(a^2 - x^2)^2}dx\right]$$ We can now compute each of these integrals: $$ = \frac{1}{a^4}\left[\ln |x| - \frac{1}{2a} \ln\left|\frac{a + x}{a - x}\right| + \frac{x}{a^2(a^2 - x^2)} \right] + C$$ where C is the constant of integration.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a powerful technique used to simplify complex fractions. It breaks down a complicated fraction into a sum of simpler fractions, making integration or other operations easier.
This is especially helpful when dealing with rational functions, where the numerator and denominator are both polynomials.The process starts by expressing the integrand as a sum of fractions with unknown coefficients. The aim is to match the original fraction to this new expression once multiplied by the denominator.
In our problem, we used partial fraction decomposition to split:

• \(\frac{1}{x(a^2 - x^2)^2} = \frac{A}{x} + \frac{B}{(a^2 - x^2)} + \frac{C}{(a^2 - x^2)^2}\)

Once we assumed the partial fraction form, we removed the denominators by equating them. This gave us a polynomial equation to solve for the unknowns A, B, and C.
Setting strategic values for \(x\), like \(x = 0, x = a\), and \(x = -a\), allowed us to solve the system of equations and find A, B, and C.
This method is very effective and often used when the denominator includes a quadratic polynomial.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two. It takes the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratics are characterized by their parabolic shapes on a graph.In this exercise, we encountered a quadratic polynomial in the denominator:

• \(a^2 - x^2\)

This polynomial is interesting because it can often be factored or further decomposed during integration. Despite the complexity it initially presents, the quadratic form allows for various techniques such as partial fraction decomposition or completing the square.
Quadratics are essential in calculus as they frequently appear in integral and differential equations. Recognizing and manipulating quadratics, like rearranging \(a^2 - x^2\), can lead to simpler solutions and reveal more about the integrand's behavior.

Computer Algebra System

A computer algebra system (CAS) assists in performing mathematical computations symbolically rather than numerically. These systems are invaluable tools for mathematicians and engineers, as they simplify complex symbolic calculations like derivatives and integrals.In our exercise, a CAS can automatically perform rather tedious algebraic manipulations such as partial fraction decomposition. It efficiently handles intricate expressions to help reach solutions much faster than manual calculations would.
Using a CAS, complex indefinite integrals such as our example:

• \(\int \frac{d x}{x\left(a^{2}-x^{2}\right)^{2}}\)

can be evaluated with ease. CAS tools like Mathematica, Maple, or even online tools provide step-by-step solutions and illustrations to deepen understanding.
By automating calculations, a CAS frees up mental resources. This allows students to focus more on understanding concepts rather than getting bogged down in arithmetic details.