Question

Partial Fractions with Repeated Linear Factors If \(f(x)=\frac{P(x)}{(x-r)^{m}}\) is a rational function with the degree of \(P\) less than \(m,\) then the partial fraction decomposition of \(f\) is \(f(x)=\frac{A_{1}}{x-r}+\frac{A_{2}}{(x-r)^{2}}+\ldots+\frac{A_{m}}{(x-r)^{m}}\) For example, \(\frac{4 x}{(x-2)^{2}}=\frac{4}{x-2}+\frac{8}{(x-2)^{2}}\) Use partial fractions to find the following integrals: (a) \(\int \frac{5 x}{(x+3)^{2}} d x\) (b) \(\int \frac{5 x}{(x+3)^{3}} d x \quad(\) Hint: Use part (a).)

Short answer

The integrals for part (a) and (b) are -5ln|x+3| - 5/(x+3) + C and -5ln|x+3| - 5/(2(x+3)^2) + 5/(x+3) + C respectively.

Step-by-step solution

Decompose the given function into simpler fractions

(a) Given function is \(\frac{5 x}{(x+3)^{2}}\). This is already into the desired format with a degree of denominator (2) greater than the numerator (1). So, the partial fraction is: \(\frac{5x}{(x+3)^2} = \frac{A}{x+3} + \frac{B}{(x+3)^2} \). (b) Our given function is \(\frac{5 x}{(x+3)^{3}}\). The partial fraction decomposition would be \(\frac{5x}{(x+3)^3} = \frac{A}{x+3} + \frac{B}{(x+3)^2} + \frac{C}{(x+3)^3} \).

Calculate constants A and B for part (a)

Comparing the coefficients on both sides, calculate the values of A and B. Multiply both sides by \((x+3)^2\) and then substituting x with -3 will give us B = 5. For A, choose another suitable value for x and solve for A. If we take x = 0, then A = -5.

Integrate the decomposed function from Step 1

Now that we've found A and B, we can rewrite our integral and then integrate: \(\int \frac{5x}{(x+3)^2}\) dx = \(\int (\frac{-5}{x+3}+\frac{5}{(x+3)^2})\) dx = -5 \(\int \frac{1}{x+3} dx\) + 5 \(\int \frac{1}{(x+3)^2} dx\). After integration, we get -5ln|x+3| - 5/(x+3) + C, where C is the constant of integration.

Apply same steps to part (b) using the hints

Using the hint to use part (a) for part (b), after performing the integration in a similar way, we get -5ln|x+3| - 5/(2(x+3)^2) + 5/(x+3) + C for part (b).

Key concepts

Repeated Linear Factors

In calculus, particularly when dealing with integration techniques, we often come across rational functions that need to be integrated. A common scenario is when these functions contain repeated linear factors in their denominators. A repeated linear factor is simply a factor of the form \( (x - r)^m \) where \( r \) is a real number, and \( m \) is a positive integer indicating the multiplicity of the root \( r \).

To integrate such functions, partial fraction decomposition is a valuable technique. It allows us to express a complicated rational function as a sum of simpler fractions whose denominators are powers of \( (x - r) \) up to \( m \) times. Each term in this decomposition has its own coefficient, denoted by \( A_k \) for \( k = 1, 2, \) …, \( m \). Once the decomposition is determined, integration becomes more manageable since we can integrate term by term.

Integrals of Rational Functions

The integration of rational functions, which are quotients of polynomials, can be complex due to their denominators. However, by using partial fraction decomposition, we can split these functions into components with simpler denominators, making them easier to integrate. This process involves finding the integral of each component:

For example, to integrate \( \frac{5x}{(x+3)^2} \), we first decompose the function into \( \frac{A}{x+3} + \frac{B}{(x+3)^2} \), where \( A \) and \( B \) are constants to be found. Each component can be readily integrated using basic integration rules or techniques like the power rule. After determining the coefficients and decomposing the function, we integrate each term to find the antiderivative of the original function.

Calculus

Calculus is a branch of mathematics that deals with derivatives and integrals of functions. In the context of integration, the goal is to find the antiderivative or the area under the curve of the function being integrated. Understanding how to integrate various functions, particularly rational functions, is crucial. Not only does this allow students to solve complex real-world problems, but it also forms the foundation for more advanced studies in mathematics and physics.

Determining the integral of a function often requires a series of steps and can involve a range of techniques. For instance, integrating functions with repeated linear factors cannot be done directly, and here is where the method of partial fraction decomposition becomes a powerful tool. This method is part of the larger set of skills that students learn in calculus to handle a variety of integrals.

Integration Techniques

There are several integration techniques that are used to evaluate integrals that are not immediately solvable by basic methods. These include substitution, integration by parts, trigonometric integrals, trigonometric substitutions, and partial fraction decomposition, among others.

Partial fraction decomposition is especially useful for integrating rational functions, where you express a fraction as a sum of simpler fractions before integration. It simplifies the original integral into components that are easier to manage. For integrals involving repeated linear factors, after finding the appropriate decomposition, one can often use a direct antiderivative formula for each term. This is not just a technique to make integrals solvable; it's a concept that shows the beauty and interconnectedness of algebra and calculus in tackling problems involving rational functions.