Question

Set up the partial fraction decomposition for \(\int \frac{x+2}{(x+3)^{3}(x-4)^{2}} d x .\) (Do not solve for the coefficients or complete the integration.)

Short answer

Write the decomposition as: \(\frac{A}{x+3}+\frac{B}{(x+3)^2}+\frac{C}{(x+3)^3}+\frac{D}{x-4}+\frac{E}{(x-4)^2}\).

Step-by-step solution

Identify the form of the denominator

The given denominator is \((x+3)^3(x-4)^2\). It is composed of distinct linear factors \((x+3)\) and \((x-4)\), with powers 3 and 2, respectively.

Set up the partial fraction decomposition structure

For the factor \((x+3)^3\), include fractions of the forms \(\frac{A}{x+3}\), \(\frac{B}{(x+3)^2}\), and \(\frac{C}{(x+3)^3}\). For the factor \((x-4)^2\), include fractions of the forms \(\frac{D}{x-4}\) and \(\frac{E}{(x-4)^2}\).

Write the complete partial fraction decomposition

Combine the terms from each factor to form the partial fraction decomposition:\[\frac{x+2}{(x+3)^3(x-4)^2} = \frac{A}{x+3} + \frac{B}{(x+3)^2} + \frac{C}{(x+3)^3} + \frac{D}{x-4} + \frac{E}{(x-4)^2}.\]

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics concerned with the accumulation of quantities and the areas under and between curves. It is one of the two main branches of calculus, alongside differential calculus. The core idea of integral calculus is the concept of an integral. An integral is a mathematical object that can be interpreted as the area under a curve. There are two main types of integrals:

• Definite Integrals: These calculate the accumulated quantity, such as area, between defined limits or bounds. A definite integral has a start and an end point.
• Indefinite Integrals: These represent a family of functions and are used to find the general form of antiderivatives. An indefinite integral does not have specified limits.

In the context of rational functions, integration often involves breaking down complex expressions into simpler parts to facilitate the calculation. This process can make use of techniques such as partial fraction decomposition, especially when faced with rational functions.

Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). They are a crucial component in integral calculus because many integrals involve rational functions. These functions are called 'rational' because they are ratios of polynomials.
When faced with the task of integrating a rational function, the goal is often to simplify the function into a form that is easier to integrate. This is where techniques like partial fraction decomposition come into play.

• Simplification: Before integrating, simplifying the polynomial fraction by canceling out common factors or factoring polynomials can make the integration process easier.
• Decomposition: Expressing the rational function as a sum of simpler fractions makes finding the integral more feasible.

Understanding the structure of rational functions is key to successfully applying these methods.

Fraction Decomposition

Fraction decomposition, particularly partial fraction decomposition, is a method used to express a fraction as a sum of simpler fractions. It is especially useful when dealing with the integration of rational functions. This technique transforms the integrand into a form that is easier to deal with.
In your exercise, the decomposition looks like this: \( \frac{x+2}{(x+3)^3(x-4)^2} \). To decompose the fraction, analyze the denominator, which consists of factors \((x+3)^3\) and \((x-4)^2\). Each factor is treated separately:

• For \((x+3)^3\): Write fractions \(\frac{A}{x+3}\), \(\frac{B}{(x+3)^2}\), and \(\frac{C}{(x+3)^3}\) to account for each power.
• For \((x-4)^2\): Write fractions \(\frac{D}{x-4}\) and \(\frac{E}{(x-4)^2}\).

By addressing each factor individually, decompose the fraction step by step, which permits easier integration.

Integration Techniques

Integration techniques are strategies and methods used to solve integrals, simplifying the integration process. A variety of techniques exist to handle different types of integrands, especially when dealing with complex rational functions.

Some common techniques include:

• Substitution: A method often used when an integral contains a composite function, where one substitutes \( u = g(x) \) and replaces \( dx \) with \( du \).
• Integration by Parts: Best used when the product of functions is challenging to integrate directly. Based on the product rule for differentiation, it's represented by \( \int u \, dv = uv - \int v \, du \).
• Partial Fraction Decomposition: Particularly useful for rational functions, breaking the fraction down into simpler, easily integrable fractions.

Using these techniques, especially partial fraction decomposition, helps solve integrals that otherwise might seem insurmountable. Each method leverages a unique approach to simplify the steps to reaching a solution.