Integration Of Sin 2x X 2

The integral of (\sin(2x) \cdot x^2) presents a classic challenge in calculus, combining trigonometric functions with polynomial terms. This specific form requires the strategic application of integration by parts, a fundamental technique for products of functions. Understanding this process not only solves this particular problem but also builds a versatile skill applicable to numerous similar integrals encountered in physics, engineering, and advanced mathematics. Let’s break down the solution step by step.

Introduction

The integral (\int \sin(2x) \cdot x^2 dx) involves multiplying a trigonometric function, (\sin(2x)), with a quadratic polynomial, (x^2). Direct integration isn’t feasible due to the product nature of the terms. Instead, we employ integration by parts, derived from the product rule for differentiation. This method transforms the integral into a potentially simpler form. The formula is: [ \int u dv = uv - \int v du ] We must strategically choose (u) and (dv) to simplify the resulting integral. A common heuristic is to select (u) as the polynomial term (which simplifies upon differentiation) and (dv) as the trigonometric term (which has a known antiderivative).

Steps for Integration by Parts

Step 1: Initial Setup

Choose:

• (u = x^2) (polynomial term)
• (dv = \sin(2x) dx) (trigonometric term)

Compute (du) and (v):

• (du = 2x dx) (derivative of (x^2))
• (v = \int \sin(2x) dx = -\frac{1}{2} \cos(2x)) (antiderivative of (\sin(2x)))

Step 2: Apply Integration by Parts Formula

Substitute into the formula: [ \int \sin(2x) \cdot x^2 dx = x^2 \cdot \left(-\frac{1}{2} \cos(2x)\right) - \int \left(-\frac{1}{2} \cos(2x)\right) \cdot 2x dx ] Simplify: [ = -\frac{x^2}{2} \cos(2x) + \int x \cos(2x) dx ]

Step 3: Handle the New Integral

Now solve (\int x \cos(2x) dx), which again requires integration by parts. Choose:

• (u = x)
• (dv = \cos(2x) dx)

Compute (du) and (v):

• (du = dx)
• (v = \int \cos(2x) dx = \frac{1}{2} \sin(2x))

Step 4: Apply Integration by Parts Again

Substitute: [ \int x \cos(2x) dx = x \cdot \left(\frac{1}{2} \sin(2x)\right) - \int \left(\frac{1}{2} \sin(2x)\right) \cdot dx ] Simplify: [ = \frac{x}{2} \sin(2x) - \frac{1}{2} \int \sin(2x) dx ]

Step 5: Integrate the Remaining Trigonometric Term

[ \int \sin(2x) dx = -\frac{1}{2} \cos(2x) + C ] Thus: [ \int x \cos(2x) dx = \frac{x}{2} \sin(2x) - \frac{1}{2} \left(-\frac{1}{2} \cos(2x)\right) + C = \frac{x}{2} \sin(2x) + \frac{1}{4} \cos(2x) + C ]

Step 6: Combine Results

Substitute back into the original equation: [ \int \sin(2x) \cdot x^2 dx = -\frac{x^2}{2} \cos(2x) + \left( \frac{x}{2} \sin(2x) + \frac{1}{4} \cos(2x) \right) + C ] Combine like terms: [ = -\frac{x^2}{2} \cos(2x) + \frac{x}{2} \sin(2x) + \frac{1}{4} \cos(2x) + C ] Factor out (\cos(2x)): [ = \cos(2x) \left( -\frac{x^2}{2} + \frac{1}{4} \right) + \frac{x}{2} \sin(2x) + C ] Or: [ = -\frac{x^2}{2} \cos(2x) + \frac{x}{2} \sin(2x) - \frac{1}{4} \cos(2x) + C ]

Scientific Explanation

This result arises from the systematic application of integration by parts twice. The polynomial term (x^2) simplifies significantly upon differentiation (to (2x)), while the trigonometric term (\sin(2x)) has a well-known antiderivative. Each application reduces the polynomial degree, making subsequent steps manageable. The process highlights the interplay between polynomial growth and trigonometric oscillation, a common theme in oscillatory systems described by differential equations. The final expression combines polynomial, sine, and cosine terms, reflecting the integral's dependence on both algebraic and periodic behavior.

Frequently Asked Questions (FAQ)

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Frequently Asked Questions (FAQ)

Q1: Why is it necessary to apply integration by parts twice in this problem?
A: The integrand is a product of a polynomial of degree 2 and a trigonometric function. After the first application, the polynomial’s degree drops to 1, but a new product of a linear term and a cosine emerges. Since that product still mixes algebraic and trigonometric factors, a second integration‑by‑parts step is required to eliminate the remaining polynomial factor.

Q2: Could a different choice of (u) and (dv) simplify the process?
A: Yes. Selecting (u = \sin(2x)) and (dv = x^{2},dx) would generate a more complicated antiderivative for (v) (a cubic polynomial) and would not reduce the difficulty. The strategy of differentiating the polynomial and integrating the trigonometric factor is the most efficient route because each differentiation lowers the polynomial’s degree, eventually reaching a constant that can be integrated directly.

Q3: What role does the constant of integration (C) play in the final expression?
A: The constant (C) represents the family of antiderivatives that differ only by an additive constant. Since indefinite integrals describe a set of functions whose derivatives equal the original integrand, any valid antiderivative can be shifted upward or downward by a constant without affecting the derivative.

Q4: How does this method generalize to integrals of the form (x^{n}\sin(ax)) or (x^{n}\cos(ax))?
A: The same iterative integration‑by‑parts approach works for any non‑negative integer (n). Each application reduces the exponent (n) by one, eventually producing a term that integrates to a simple sinusoid. After (n) steps, the remaining integral is a constant‑multiple of (\sin(ax)) or (\cos(ax)), which can be integrated directly.

Q5: Are there alternative techniques that bypass repeated integration by parts?
A: Yes. One can use complex exponentials, writing (\sin(2x)=\Im(e^{i2x})) and integrating (x^{2}e^{i2x}) via repeated differentiation under the integral sign, or employ tabular integration (a shortcut version of integration by parts) that records successive derivatives and antiderivatives in a table. Both methods arrive at the same result but may be more convenient for higher‑order polynomials.

Conclusion

The integral (\displaystyle\int x^{2}\sin(2x),dx) exemplifies how integration by parts can systematically dismantle products of polynomials and trigonometric functions. By strategically differentiating the algebraic factor and integrating the trigonometric factor, each iteration reduces the problem’s complexity until elementary integrals remain. The final antiderivative, [ -\frac{x^{2}}{2}\cos(2x)+\frac{x}{2}\sin(2x)-\frac{1}{4}\cos(2x)+C, ] encapsulates the interplay between polynomial growth and oscillatory behavior. This methodology not only provides a concrete solution but also serves as a template for tackling a broader class of integrals that arise in physics, engineering, and applied mathematics, where polynomial amplitudes modulate periodic phenomena.

The integral (\int x^{2}\sin(2x),dx) serves as a classic illustration of integration by parts, demonstrating how the method systematically reduces complexity when dealing with products of polynomials and trigonometric functions. By differentiating the polynomial factor and integrating the trigonometric factor, each iteration lowers the degree of the polynomial until only elementary integrals remain. This process not only yields the specific antiderivative [ -\frac{x^{2}}{2}\cos(2x)+\frac{x}{2}\sin(2x)-\frac{1}{4}\cos(2x)+C ] but also exemplifies a general strategy applicable to integrals of the form (x^{n}\sin(ax)) or (x^{n}\cos(ax)). The constant (C) underscores that indefinite integrals represent families of functions differing by a vertical shift, reflecting the inherent non-uniqueness of antiderivatives.

Alternative approaches, such as using complex exponentials or tabular integration, can streamline the computation, especially for higher powers of (x), but the underlying principle remains the same: decompose the integrand into manageable pieces and recombine them through systematic application of integration by parts. This methodology is widely relevant in physics and engineering, where polynomial amplitudes modulate periodic phenomena, and mastering it equips one to handle a broad class of problems with confidence and efficiency.

Continuing the discussion of theintegral (\int x^{2}\sin(2x),dx), the systematic application of integration by parts reveals not only the specific antiderivative but also underscores the power and versatility of the method for a wide class of problems. The result, (-\frac{x^{2}}{2}\cos(2x) + \frac{x}{2}\sin(2x) - \frac{1}{4}\cos(2x) + C), is a testament to the careful orchestration of differentiation and integration. This process systematically dismantles the polynomial factor, reducing its degree step-by-step, while the trigonometric factor provides the necessary oscillation, ensuring the product's complexity is managed.

The elegance of this approach lies in its adaptability. While the specific steps for (x^2) are clear, the same principle extends seamlessly to higher powers of (x), such as (x^n\sin(ax)) or (x^n\cos(ax)). Each additional factor of (x) simply requires one more iteration of integration by parts, a predictable pattern that becomes intuitive with practice. This scalability is crucial for tackling more complex integrals encountered in advanced calculus, physics, and engineering.

Moreover, the result itself is significant. The presence of both polynomial terms ((x^2), (x)) and trigonometric terms ((\cos(2x)), (\sin(2x))) reflects the fundamental nature of the integrand. The polynomial terms represent the "amplitude" modulation of the oscillatory behavior, a common theme in wave phenomena, signal processing, and harmonic analysis. Understanding the exact form of the antiderivative provides insight into the system's response, a vital consideration in modeling real-world physical systems.

While the detailed tabular method or complex exponential approaches offer computational shortcuts for specific cases, particularly with higher powers, they ultimately rely on the same core principle: decomposing the integrand into simpler components whose integrals are known, and then systematically recombining them using the rules of integration by parts. This foundational understanding is indispensable, as it provides the conceptual framework that makes these shortcuts meaningful and applicable.

Therefore, mastering the technique demonstrated here for (\int x^{2}\sin(2x),dx) is far more than solving a single integral. It equips the practitioner with a robust, systematic strategy for confronting a vast array of integrals involving products of polynomials and trigonometric or exponential functions. This skill is not merely academic; it is a practical tool essential for analyzing and solving problems across numerous scientific and engineering disciplines where oscillatory behavior is modulated by polynomial factors. The ability to derive and manipulate such antiderivatives forms a cornerstone of advanced mathematical problem-solving.

Conclusion

The systematic application of integration by parts to the integral (\int x^{2}\sin(2x),dx) yields a specific antiderivative and exemplifies a powerful, general methodology. This approach, by strategically differentiating the polynomial factor and integrating the trigonometric factor, systematically reduces the problem's complexity, transforming a product of polynomial and trigonometric functions into a combination of elementary integrals. The resulting antiderivative, (-\frac{x^{2}}{2}\cos(2x) + \frac{x}{2}\sin(2x) - \frac{1}{4}\cos(2x) + C), captures the interplay between polynomial growth and oscillatory behavior inherent in the integrand.

This technique is not confined to this specific case. Its principles extend readily to integrals of the form (x^n \sin(ax)) or (x^n \cos(ax)) for any positive integer (n), providing a scalable framework for tackling increasingly complex products. The adaptability of integration by parts makes it an indispensable tool in the mathematician's and engineer's toolkit.

While alternative methods like tabular integration or complex exponentials offer computational efficiency, particularly for higher powers, they are applications of the underlying principle of decomposition and systematic recomposition. Understanding the core process is fundamental to leveraging these shortcuts effectively.

Ultimately, the solution to (\int x^{

(2\sin(2x),dx) represents a stepping stone to a deeper comprehension of integral techniques. It’s a demonstration of how a seemingly daunting problem can be broken down into manageable pieces, revealing the elegance and power of mathematical manipulation. The ability to recognize and apply integration by parts – and to understand why it works – is a skill that transcends isolated problems, fostering a more intuitive and confident approach to tackling a wide range of integral challenges. It’s a testament to the idea that mathematical proficiency isn’t simply about memorizing formulas, but about grasping the underlying logic and applying it creatively. Therefore, the successful resolution of this integral serves not just as a numerical answer, but as a foundational lesson in the art and science of integration.