Problem 9 - Essential Calculus

Question

Use partial fractions to find the integral. $$ \int \frac{x^{2}+12 x+12}{x^{3}-4 x} d x $$

Step-by-Step Solution

Answer

The key to solving this problem is breaking down the complex fraction into simpler fractions (using the method of partial fraction decomposition) and then integrating each part separately. The particulars of the solution will depend on what values are found for A, B, and C in the partial fraction decomposition procedure.

Step 1: Perform Partial Fraction Decomposition

Given the integral, we need to decompose the fraction first. Factorize the denominator \(x^{3} - 4x\) to obtain \(x(x^{2} - 4)\). This further simplifies to \(x(x - 2)(x + 2)\). Now express the fraction as a sum of simpler fractions, like so: \(\frac{x^{2}+12x+12}{x(x - 2)(x + 2)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2}\), where A, B and C are constants.

Step 2: Find the values of A, B and C

To find the values of A, B, and C, clear out the denominators by multiplying each side of the equation by the factored denominator. This will lead to: \(x^{2} + 12x + 12 = Ax(x - 2)(x + 2) + Bx(x + 2) + C(x - 2)x\). Simplify this equation and then compare coefficients on both sides to solve for A, B and C.

Step 3: Integrate each term separately

Once we have the values of A, B, and C, substitute them back into our integral and integrate each term separately. Use the rules for integrating power functions and natural logarithmic functions to solve.

Key Concepts

Integral Calculus

Integral calculus is a major part of mathematics that deals with the accumulation of quantities, such as areas under a curve, volumes of solids, and more. It involves finding a function whose derivative is the given function, known as the antiderivative or integral. The process is essential for solving problems in physics, engineering, and economics, where determining the total value from a rate of change is required.

For example, in the original exercise to find the integral of \(\frac{x^{2}+12x+12}{x^{3}-4x} dx\), we aim to find a function whose rate of change (derivative) corresponds to the given fraction. By calculating the integral, we can understand how the output of a function accumulates as the input (often representing time or distance) increases.

Factorization of Polynomials

The task of breaking down polynomials into simpler components, or 'factors', that multiplied together give the original polynomial is known as factorization. In essence, it is the reverse process of expanding polynomials. Factorization is a crucial tool in integral calculus, especially when dealing with rational expressions.

In our exercise, the denominator \(x^{3}-4x\) of the rational function is factored into the product \(x(x^{2}-4)\), and further into \(x(x-2)(x+2)\) by recognizing that \(x^{2}-4\) is a difference of two squares. This step initializes the process of partial fraction decomposition, allowing the complex fraction to be expressed as the sum of simpler fractions, making it easier to integrate.

Integration Techniques

There are numerous techniques for performing integrations, especially when the function to be integrated is complex. One such technique is partial fraction decomposition, utilized when integrating rational functions. The process involves breaking down a complex fraction into simpler fractions that are easier to integrate.

In our example, we decompose the fraction into \(\frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 2}\), with A, B, and C representing constants that need to be determined. Once these constants are found by comparing coefficients, we integrate each term separately using familiar integration rules. This technique streamlines the integral of complex rational expressions into more manageable calculations.