Problem 5 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{x-12}{x^{2}-4 x} d x$$

Step-by-Step Solution

Answer

The solution to the integral \(\int \frac{x-12}{x^{2}-4 x} d x\) is \(4 \ln|x| - \ln|x-4| + C\)

Step 1: Factorize the denominator

Factorize the denominator \(x^{2}-4 x\) as \(x(x-4)\)

Step 2: Partial Fraction Decomposition

Rewrite the fraction \(\frac{x-12}{x^{2}-4 x}\) as \(\frac{A}{x} + \frac{B}{x-4}\) where A and B are constants to be determined. Solving for A and B gives us A = 4 and B = -1.

Step 3: Integrate the decomposed fraction

Integrate \( \int \frac{4}{x} dx - \int \frac{1}{x-4} dx \) which gives us \(4 \ln|x| - \ln|x-4| + C\)

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique for breaking down complex fractions into simpler parts, making integration easier. When dealing with rational functions, ones where you have a polynomial divided by another polynomial, this method is invaluable. The goal is to express the fraction as a sum of simpler fractions.
For example, in the exercise provided, the fraction \( \frac{x-12}{x^{2}-4x} \) was decomposed into the form \( \frac{A}{x} + \frac{B}{x-4} \).

• The denominator, \( x^2 - 4x \), was first factorized into \( x(x-4) \).
• We then expressed the original fraction in terms of these factors.
• A and B are constants that were solved by plugging the values that nullify terms of each fraction, leading us to determin A as 4 and B as -1.

This decomposition simplifies the integration process significantly, turning a tricky integral into something manageable.

Definite and Indefinite Integrals

Integrals can be categorized as either definite or indefinite, depending on whether they calculate the area under a curve or yield a family of functions. Indefinite integrals, like the one in the step-by-step solution, find the antiderivative of a function. This is represented with a constant \( C \) because any of a range of constants can be added to an antiderivative.
In the provided example, after integrating, we end up with \( 4 \ln|x| - \ln|x-4| + C \). Here, \( C \) signifies that an infinite number of antiderivatives exist, each differing by a constant.

• Indefinite integrals do not have limits.
• They represent a large set of possible solutions.
• The symbol \( \int \) indicates the integration process, and we add \( dx \) to denote integration with respect to \( x \).

In contrast, definite integrals evaluate to a specific value and have upper and lower limits, marking the interval on the x-axis over which the area is calculated.

Integration Techniques

Integration techniques are methods that help us calculate integrals, especially when dealing with complex functions. Among these techniques, partial fraction decomposition is a common method for integrating rational functions. Here are some typical techniques that simplify this process:

• **Substitution:** Used when a substitution can simplify the integral, making it easier to evaluate.
• **Integration by Parts:** This method is useful when the integrand is a product of two functions, applying the formula \( \int u \, dv = uv - \int v \, du \).
• **Trigonometric Identities:** These can simplify integrals involving trigonometric functions by rewriting them in a different form.
• **Partial Fraction Decomposition:** As previously discussed, this method helps when dealing with rational functions, simplifying them into more manageable pieces.

In the presented exercise, partial fraction decomposition was the key technique that streamlined the integration process, turning a difficult task into an approachable one.