Solving The Integral ∫ [x^2(x^2 + 2)]/[(x^2 + 1)(x Cos X - Sin X)^2] Dx A Step-by-Step Guide

Navigating the world of integral calculus can often feel like traversing a complex labyrinth, with each integral presenting its unique set of challenges and requiring a tailored approach. Among these intriguing mathematical puzzles, the integral ∫[x^2(x2 + 2)]/[(x^2 + 1)(x cos x - sin x)^2] dx stands out as a particularly captivating specimen. This article aims to dissect this integral, offering a step-by-step guide to its solution while illuminating the underlying mathematical principles and techniques involved. Our exploration will not only provide a solution but also delve into the thought process behind choosing the right integration strategies, making it a valuable resource for students, educators, and anyone with a passion for calculus.

Understanding the Integral's Structure

Before diving into the solution, it's crucial to understand the integral's structure. The integral ∫[x^2(x2 + 2)]/[(x^2 + 1)(x cos x - sin x)^2] dx presents a rational function multiplied by a trigonometric component, making it a non-trivial problem. The numerator x^2(x2 + 2) and the denominator (x^2 + 1)(x cos x - sin x)^2 suggest that a direct substitution might not be immediately obvious. The presence of the (x cos x - sin x)^2 term hints at the possibility of a clever substitution or a trigonometric manipulation strategy. This initial assessment is crucial in charting our course through the solution process.

Breaking Down the Components

To effectively tackle this integral, let's dissect its key components: the numerator, x^2(x2 + 2), which is a polynomial of degree four; the term (x^2 + 1) in the denominator, a simple quadratic; and the trigonometric expression (x cos x - sin x)^2, which is the square of a function involving both algebraic and trigonometric elements. Recognizing these components is the first step towards identifying potential strategies for simplification and integration. The interplay between the polynomial and trigonometric parts suggests exploring techniques that can exploit their relationship, such as differentiation or substitution that simplifies the trigonometric part while also dealing with the polynomial.

The Role of (x cos x - sin x)^2

At the heart of the integral lies the term (x cos x - sin x)^2. This term is not just a trigonometric function; it's a composite function that holds the key to simplifying the entire integral. The derivative of (x cos x - sin x) is -x sin x, which might seem unrelated at first glance. However, it suggests that a substitution involving the quotient rule could be a viable strategy. By recognizing this, we can start to form a plan for how to approach the integral more systematically. This specific term often dictates the path to the solution, emphasizing the importance of careful observation and strategic foresight in integral calculus.

The Strategic Substitution: Unveiling the Solution

The most effective approach to this integral lies in recognizing a strategic substitution that simplifies the complex trigonometric term. The keen observer will notice that the derivative of (sin x / x) plays a pivotal role. Let's delve into this substitution and how it elegantly unravels the integral.

Identifying the Key Substitution

The key to solving this integral is the realization that the derivative of (sin x / x) is closely related to the denominator. Specifically, using the quotient rule, the derivative of (sin x / x) is (x cos x - sin x) / x^2. This observation is the cornerstone of our solution strategy. By recognizing this derivative, we can construct a substitution that dramatically simplifies the integral, transforming it into a more manageable form. This type of recognition is what separates routine integration problems from those that require a deeper understanding of calculus principles.

Performing the Substitution

Let's set u = sin x / x. Then, du = [(x cos x - sin x) / x^2] dx. This substitution allows us to rewrite the integral in terms of u. However, we need to manipulate the original integral to fit this substitution. This involves algebraic manipulation and a bit of foresight to see how the pieces will fall into place. The process is akin to solving a puzzle, where each step brings us closer to the complete picture. The beauty of this method lies in its ability to transform a seemingly intractable problem into a straightforward integration.

Rewriting the Integral

To effectively use our substitution, we need to rewrite the original integral in terms of u and du. This involves some algebraic manipulation. Notice that we can rewrite the original integral as:

∫ [x^2(x2 + 2) / (x^2 + 1)(x cos x - sin x)^2] dx = ∫ [(x^2 / (x cos x - sin x)^2) * ((x^2 + 2) / (x^2 + 1))] dx

Now, using our substitution, we can express dx in terms of du:

dx = [x^2 / (x cos x - sin x)] * du

Substituting this back into the integral, we get:

∫ [(x^2 / (x cos x - sin x)^2) * ((x^2 + 2) / (x^2 + 1))] * [x^2 / (x cos x - sin x)] du

This looks more complicated, but remember our goal is to simplify using u = sin x / x. The next step involves some clever algebra to get everything in terms of u. This process highlights the importance of algebraic proficiency in mastering integral calculus.

The Art of Algebraic Manipulation

The transition from the original integral to a form easily integrable with our substitution requires careful algebraic manipulation. This involves expressing the remaining terms in the integral in terms of u. The ability to perform these manipulations is a crucial skill in integral calculus.

Expressing x^2 in Terms of u

This is the trickiest part. We need to find a way to express (x^2 + 2) / (x^2 + 1) in terms of u = sin x / x. Notice that:

(x^2 + 2) / (x^2 + 1) = 1 + [1 / (x^2 + 1)]

So, our integral now looks like:

∫ [1 + (1 / (x^2 + 1))] * [x^4 / (x cos x - sin x)^2] dx

This form is still not directly in terms of u, but it brings us closer. The key now is to recognize how the derivative we identified earlier fits into this expression. The next step involves strategically using the substitution to simplify the integral further. This phase of the solution exemplifies the blend of algebraic and calculus techniques necessary for solving complex integrals.

Simplifying the Expression

Now, recall that du = [(x cos x - sin x) / x^2] dx. Therefore, we can rewrite the integral as:

∫ [1 + (1 / (x^2 + 1))] * (1 / [(x cos x - sin x) / x^2]2) dx = ∫ [1 + (1 / (x^2 + 1))] * (1 / u'^2) dx

Where u' represents the derivative of u with respect to x. This transformation is crucial because it starts to highlight the substitution's effectiveness in simplifying the integral. The expression now involves u' directly, making the relationship between the original integral and our substitution more apparent. The next step is to fully integrate the substitution into the equation, leading us closer to the final solution.

Completing the Integration

With the integral simplified in terms of u and du, the final steps involve integrating with respect to u and then substituting back to express the solution in terms of x. This stage combines the calculus of integration with the precision of algebraic substitution.

Integrating with Respect to u

Our integral, after the substitutions and manipulations, takes the form:

∫ [1 + (1 / (x^2 + 1))] * (1 / [(x cos x - sin x) / x^2]2) dx

Using our substitution du = [(x cos x - sin x) / x^2] dx, we can simplify the integral to:

∫ (1 + 1/(x^2 + 1)) * (1/ (du/dx)^2 ) dx = ∫ (1 + 1/(x^2 + 1)) * (1/ ((x cos x - sin x) / x²⁾2 ) dx

Which simplifies to:

∫ (1 + 1/(x^2 + 1)) * (x^4 / (x cos x - sin x)^2 ) dx

Now, we substitute u = sin(x)/x, then du = (x cos(x) - sin(x))/x^2 dx. We can rewrite the integral in terms of u. Let's divide the numerator and denominator of the original integrand by x^4:

∫ ((1 + 2/x^2) / ((1 + 1/x^2) * ((x cos x - sin x)/x²⁾2)) dx

Now, substitute du = (x cos x - sin x)/x^2 dx. The integral transforms into:

∫ (1 + 1/(x^2 + 1)) * (x^4 / (x cos x - sin x)^2) dx = ∫ (1 + 1/(x^2 + 1)) * (1/u'^2) dx

This leads to ∫ (1 + 1/(x^2 + 1)) * (x^4/(x cos x - sin x)^2) dx, which simplifies to:

-x / (x^2 + 1) + C

Substituting Back

Finally, we substitute u = sin x / x back into the solution to express the integral in terms of x. This step is essential for providing the answer in the original variable, completing the solution process. The final answer to the integral ∫ [x^2(x2 + 2) / (x^2 + 1)(x cos x - sin x)^2] dx is:

-x / (x^2 + 1) + C

Where C is the constant of integration. This constant is a crucial part of the solution, reminding us that the indefinite integral represents a family of functions, not just a single function. The process of substituting back highlights the importance of keeping track of our substitutions and ensuring that the final answer is in the original variables.

Conclusion: Mastering the Art of Integration

Evaluating the integral ∫ [x^2(x2 + 2) / (x^2 + 1)(x cos x - sin x)^2] dx is a journey through the core techniques of integral calculus. From recognizing the crucial substitution to performing algebraic manipulations and completing the integration, each step requires a blend of skill, insight, and perseverance. This exercise not only provides a solution to a specific problem but also enhances our understanding of the broader landscape of integral calculus. By dissecting complex integrals like this, we sharpen our analytical abilities and deepen our appreciation for the elegance and power of mathematical problem-solving. The satisfaction of arriving at the final solution is a testament to the effort invested and a motivation to tackle even more challenging mathematical puzzles.