Problem 22 - Calculus for Biology and Medicine

Question

In Problems , use partial-fraction decompositon to evaluate each integral. $$ \int \frac{3 x^{2}+4 x+3}{\left(x^{2}+1\right)^{2}} d x $$

Step-by-Step Solution

Answer

Use partial fraction decomposition to rewrite and integrate separately.

Step 1: Identify Partial Fractions

First, we need to set up the partial fraction decomposition. Since the denominator \((x^2 + 1)^2\) is a non-reducible quadratic repeated factor, the decomposition has the form \(\frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2}\).

Step 2: Construct Equation

Multiply the entire equation by the common denominator \((x^2 + 1)^2\) to find:\[3x^2 + 4x + 3 = (Ax + B)(x^2 + 1) + (Cx + D)\]Expand the right side and equate coefficients with the left side.

Step 3: Equate and Solve Coefficients

Expand the terms to give:\[3x^2 + 4x + 3 = Ax^3 + Bx^2 + Ax + B + Cx + D\]Combine terms and equate coefficients for each power of \(x\). Solve the system of equations to find \(A\), \(B\), \(C\), and \(D\).

Step 4: Integrate Each Term

Once the coefficients are determined, rewrite the integral with these decomposed fractions. Evaluate\[\int \left( \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2} \right) dx\]as separate integrals. Use integration techniques such as substitution or recognizing standard integrals.

Key Concepts

Integral Calculus

Integral calculus is a critical branch of mathematics centered on the idea of integration, which is essentially the reverse process of differentiation. In the context of this exercise, integration is used to determine the area under a curve represented by a function. The integral of a function not only gives us this area but also allows us to solve differential equations, find lengths, volumes, and much more. When performing integration, especially with complex functions, it's necessary to understand various techniques like substitution or parts integration, which simplify the process. In this exercise, the substitution method might be deployed after performing partial fraction decomposition to deal with the complex rational function. The ultimate goal is to express the result in a cleaner, more manageable form, from which the integral can be easily calculated. Understanding integral calculus is essential for delving deeper into mathematical analysis and solving complex mathematical models.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomial expressions. They are a fundamental concept in calculus due to their frequent occurrence in various problems, including integration and differentiation. In the given exercise, the function to be integrated is a rational function where the numerator is a quadratic polynomial, and the denominator is a power of a non-reducible quadratic polynomial. Working with rational functions often involves techniques such as partial fraction decomposition, which simplifies the process of integration. By breaking down complex rational expressions into simpler parts, we can solve integrals that would otherwise be quite challenging. In this context, it's crucial to understand the behavior and characteristics of polynomials, like how they factor and interact with other terms, which aids in setting up the partial fraction. By decomposing the rational function, we can tackle each fragment separately, making integration more straightforward.

Integration Techniques

Integration techniques are strategies used to find the antiderivative of functions, especially when faced with difficulties integrating directly. Key techniques include substitution, integration by parts, and partial fraction decomposition, each suited to different scenarios or types of functions. In this exercise, partial fraction decomposition is employed to simplify a complex rational function. This technique involves expressing the integrand as a sum of simpler fractions. Each part is then integrated individually, often using simpler techniques. For instance, if we encounter a standard form such as \ \( \int \frac{1}{x^2+1} dx \ \), recognizing it as the derivative of an arctangent function simplifies the process. Moreover, substitution may be used in concert with these methods to further ease the integration process. Substitution involves replacing a complex part of a function with a single variable to simplify integration. Combining these methods effectively often results in finding the integral of complex functions more efficiently.