Solving The Integral Of X² / ((x² + 1)(3x² + 4)) A Comprehensive Guide

In this comprehensive guide, we will explore the intricacies of integral calculus by dissecting a specific problem: ${ \int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} dx }$. This integral, while seemingly complex, offers a fantastic opportunity to delve into various integration techniques, particularly the method of partial fraction decomposition. Understanding these methods is crucial not only for acing calculus exams but also for tackling real-world problems in physics, engineering, and economics where integrals frequently appear. Our journey will involve breaking down the integrand into simpler fractions, applying appropriate substitutions, and meticulously evaluating each resulting integral. We will emphasize clarity and step-by-step explanations to ensure that every reader, regardless of their background, can follow along and grasp the underlying concepts. By the end of this article, you'll not only know how to solve this particular integral but also possess a deeper appreciation for the power and elegance of integral calculus. So, let's embark on this mathematical adventure together and unravel the mysteries of integration!

Understanding the Challenge: Why Partial Fractions?

Before we dive into the solution, let's understand why this integral presents a unique challenge and why partial fraction decomposition is the ideal technique. The integrand, ${ \frac{x^2}{(x^2 + 1)(3x^2 + 4)} }$, is a rational function – a ratio of two polynomials. While some rational functions can be integrated directly using basic rules or simple substitutions, this one requires a more strategic approach. The denominator, being a product of two quadratic expressions, makes direct integration difficult. This is where partial fraction decomposition comes to the rescue. This technique allows us to break down the complex rational function into a sum of simpler fractions, each of which can be integrated more easily. The core idea is to express the given fraction as a sum of fractions with denominators that are factors of the original denominator. In our case, we aim to decompose ${ \frac{x^2}{(x^2 + 1)(3x^2 + 4)} }$ into the form ${ \frac{A}{x^2 + 1} + \frac{B}{3x^2 + 4} }$, where A and B are constants that we need to determine. Once we find these constants, we can integrate each term separately, making the entire process much more manageable. This method is a cornerstone of integral calculus, and mastering it will significantly enhance your problem-solving capabilities.

Step-by-Step Solution: Decomposing the Integrand

Now, let's embark on the step-by-step solution, starting with the crucial process of partial fraction decomposition. Our goal is to express the integrand, ${ \frac{x^2}{(x^2 + 1)(3x^2 + 4)} }$, in the form ${ \frac{A}{x^2 + 1} + \frac{B}{3x^2 + 4} }$. To achieve this, we first equate the two expressions: ${ \frac{x^2}{(x^2 + 1)(3x^2 + 4)} = \frac{A}{x^2 + 1} + \frac{B}{3x^2 + 4} }$. Next, we clear the denominators by multiplying both sides of the equation by ${ (x^2 + 1)(3x^2 + 4) }$. This gives us: ${ x^2 = A(3x^2 + 4) + B(x^2 + 1) }$. Now, we need to solve for the constants A and B. There are two common methods for doing this: substituting specific values of x or equating coefficients. Let's use the method of equating coefficients. Expanding the right side of the equation, we get: ${ x^2 = 3Ax^2 + 4A + Bx^2 + B }$. Grouping like terms, we have: ${ x^2 = (3A + B)x^2 + (4A + B) }$. For this equation to hold true for all values of x, the coefficients of the corresponding powers of x must be equal. This gives us a system of two linear equations: ${ 3A + B = 1 }$ (coefficient of ${ x^2 }$) and ${ 4A + B = 0 }$ (constant term). Solving this system of equations will provide us with the values of A and B, which are the key to decomposing our integrand.

Solving for A and B: Unveiling the Constants

To solve the system of equations ${ 3A + B = 1 }$ and ${ 4A + B = 0 }$, we can use various methods, such as substitution or elimination. Let's use the elimination method, which is particularly efficient in this case. We can subtract the first equation from the second equation to eliminate B: ${ (4A + B) - (3A + B) = 0 - 1 }$. This simplifies to: ${ A = -1 }$. Now that we have found A, we can substitute it back into either of the original equations to solve for B. Let's substitute A = -1 into the first equation: ${ 3(-1) + B = 1 }$. This gives us: ${ -3 + B = 1 }$, and therefore, ${ B = 4 }$. So, we have determined that A = -1 and B = 4. This means we can now rewrite our integrand using these values in the partial fraction decomposition: ${ \frac{x^2}{(x^2 + 1)(3x^2 + 4)} = \frac{-1}{x^2 + 1} + \frac{4}{3x^2 + 4} }$. This decomposition is a crucial step, as it transforms a complex rational function into a sum of simpler fractions that are much easier to integrate. With A and B in hand, we are now well-equipped to tackle the integration process itself. The next step involves integrating each term separately, which will lead us to the final solution.

Integrating the Decomposed Fractions: A Step-by-Step Guide

Now that we've successfully decomposed our integrand, the next crucial step is to integrate the resulting fractions. We have: ${ \int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} dx = \int \left( \frac{-1}{x^2 + 1} + \frac{4}{3x^2 + 4} \right) dx }$. This can be separated into two integrals: ${ \int \frac{-1}{x^2 + 1} dx + \int \frac{4}{3x^2 + 4} dx }$. Let's tackle each integral individually. The first integral, ${ \int \frac{-1}{x^2 + 1} dx }$, is a standard integral that you should recognize. It's the negative of the arctangent function: ${ \int \frac{-1}{x^2 + 1} dx = -\arctan(x) + C_1 }$, where ${ C_1 }$ is the constant of integration. The second integral, ${ \int \frac{4}{3x^2 + 4} dx }$, requires a bit more manipulation. To make it resemble a standard form, we can factor out a 4 from the denominator: ${ \int \frac{4}{3x^2 + 4} dx = 4 \int \frac{1}{3x^2 + 4} dx }$. Now, we want to rewrite the denominator in the form ${ u^2 + a^2 }$ to utilize the arctangent integral formula. We can rewrite ${ 3x^2 + 4 }$ as ${ (\sqrt{3}x)^2 + 2^2 }$. This suggests a u-substitution. Let ${ u = \sqrt{3}x }$, so ${ du = \sqrt{3} dx }$, and ${ dx = \frac{du}{\sqrt{3}} }$. Substituting these into the integral, we get: ${ 4 \int \frac{1}{u^2 + 2^2} \frac{du}{\sqrt{3}} = \frac{4}{\sqrt{3}} \int \frac{1}{u^2 + 2^2} du }$. This integral is now in the standard form for the arctangent function. In the next section, we'll evaluate this integral and combine the results to find the complete solution.

Evaluating the Arctangent Integral and Combining Results

Continuing from the previous step, we have the integral ${ \frac{4}{\sqrt{3}} \int \frac{1}{u^2 + 2^2} du }$. This is a standard integral of the form ${ \int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan(\frac{u}{a}) + C }$. In our case, ${ a = 2 }$, so the integral becomes: ${ \frac{4}{\sqrt{3}} \cdot \frac{1}{2} \arctan(\frac{u}{2}) + C_2 = \frac{2}{\sqrt{3}} \arctan(\frac{u}{2}) + C_2 }$, where ${ C_2 }$ is the constant of integration. Now, we need to substitute back for u: ${ u = \sqrt{3}x }$. So, the integral becomes: ${ \frac{2}{\sqrt{3}} \arctan(\frac{\sqrt{3}x}{2}) + C_2 }$. We now have the solutions for both individual integrals. Recall that we had: ${ \int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} dx = \int \frac{-1}{x^2 + 1} dx + \int \frac{4}{3x^2 + 4} dx }$. We found that ${ \int \frac{-1}{x^2 + 1} dx = -\arctan(x) + C_1 }$ and ${ \int \frac{4}{3x^2 + 4} dx = \frac{2}{\sqrt{3}} \arctan(\frac{\sqrt{3}x}{2}) + C_2 }$. Combining these results, we get the final solution: ${ \int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} dx = -\arctan(x) + \frac{2}{\sqrt{3}} \arctan(\frac{\sqrt{3}x}{2}) + C }$, where ${ C = C_1 + C_2 }$ is the overall constant of integration. This completes the integration process. We have successfully broken down the complex integral, applied partial fraction decomposition, and integrated each resulting term to arrive at the final solution. This methodical approach highlights the power and elegance of calculus techniques in solving seemingly daunting problems.

Conclusion: The Power of Integration Techniques

In conclusion, we have successfully navigated the complexities of integrating the rational function ${ \frac{x^2}{(x^2 + 1)(3x^2 + 4)} }$. This journey has not only provided us with the solution to this specific integral but has also illuminated the broader significance of integration techniques in calculus. The key to solving this integral lay in the application of partial fraction decomposition, a powerful method for simplifying rational functions. By breaking down the complex integrand into simpler fractions, we were able to apply standard integration formulas and substitutions more effectively. The step-by-step process, from decomposing the integrand to evaluating the individual integrals, showcased the methodical approach required to tackle challenging calculus problems. Moreover, this exercise underscores the importance of recognizing standard integral forms and strategically applying substitutions to transform integrals into solvable forms. The final solution, ${ -\arctan(x) + \frac{2}{\sqrt{3}} \arctan(\frac{\sqrt{3}x}{2}) + C }$, represents the culmination of these techniques. More broadly, the ability to solve integrals like this is fundamental in many areas of science, engineering, and economics, where integrals are used to model and solve a wide range of problems, from calculating areas and volumes to modeling physical systems and economic behavior. Mastering these techniques not only enhances your mathematical prowess but also equips you with essential tools for problem-solving in various disciplines.

This exploration of ${ \int \frac{x^2}{(x^2 + 1)(3x^2 + 4)} dx }$ serves as a testament to the beauty and utility of integral calculus. The combination of algebraic manipulation, strategic decomposition, and the application of standard integral forms demonstrates the power and elegance of mathematical problem-solving. As you continue your journey in calculus and beyond, the techniques and insights gained from this example will undoubtedly serve you well in tackling new challenges and unraveling the complexities of the mathematical world.