Evaluate The Definite Integral Of 1/(x^2+4) From 0 To 2.

In the captivating world of calculus, definite integrals stand as powerful tools for calculating areas, volumes, and a myriad of other quantities. Among the diverse landscape of integrals, some possess elegant solutions that unveil fundamental mathematical principles. This article delves into the evaluation of a specific definite integral, revealing the underlying techniques and concepts that make calculus such a fascinating field. Our mission is to meticulously dissect and conquer the following integral:

∫₀² dx/(x²+4)

This seemingly simple integral holds within it a gateway to understanding trigonometric substitution, a technique that proves invaluable when dealing with integrands involving quadratic expressions. Join us as we embark on this mathematical journey, carefully unraveling the steps required to arrive at the solution. This exploration isn't just about finding the answer; it's about appreciating the beauty and interconnectedness of mathematical ideas. As we progress, we'll emphasize the importance of each step, ensuring a clear and comprehensive understanding of the process. This problem serves as a perfect illustration of how a seemingly complex integral can be tamed with the right approach and a solid grasp of fundamental principles. By mastering techniques like trigonometric substitution, we equip ourselves with the tools to tackle a wide range of integration challenges. So, let's dive in and unlock the secrets hidden within this integral!

Unveiling the Trigonometric Substitution Technique

To effectively tackle the integral ∫₀² dx/(x²+4), we introduce a powerful technique known as trigonometric substitution. This method is particularly adept at handling integrals containing expressions in the form of a² + x², a² - x², or x² - a², where 'a' is a constant. In our case, we have the expression x² + 4, which beautifully aligns with the a² + x² pattern, with a² equaling 4 (and consequently, a equaling 2). The core idea behind trigonometric substitution is to cleverly replace the variable 'x' with a trigonometric function, thereby transforming the integral into a more manageable form. For expressions of the form a² + x², the substitution x = a tan θ is the go-to choice. This substitution is motivated by the trigonometric identity 1 + tan² θ = sec² θ, which allows us to simplify the expression under the integral. By making this strategic substitution, we aim to eliminate the square root or, in this instance, transform the denominator into a simpler expression involving trigonometric functions. This simplification is the key to unlocking the integral's solution. However, it's not merely about substitution; it's about carefully choosing the right substitution and understanding how it affects the limits of integration. This is where the finesse of calculus comes into play. We must meticulously track the changes in variables and limits to ensure we arrive at the correct final answer. The power of trigonometric substitution lies in its ability to transform seemingly intractable integrals into solvable ones. It's a testament to the elegance and ingenuity of mathematical techniques.

Applying the Substitution: A Step-by-Step Guide

Let's put our knowledge of trigonometric substitution into action and conquer the integral ∫₀² dx/(x²+4). Following the principles outlined earlier, we'll substitute x = 2 tan θ. This strategic move is the cornerstone of our approach, setting the stage for simplification and eventual solution. Now, a crucial step in any substitution method is finding the differential dx in terms of dθ. Differentiating x = 2 tan θ with respect to θ, we get dx = 2 sec² θ dθ. This differential relationship is essential for transforming the entire integral into the θ domain. But our task isn't complete yet. We must also address the limits of integration. The original limits were in terms of 'x' (from x = 0 to x = 2), but now we're integrating with respect to 'θ'. So, we need to find the corresponding θ values. When x = 0, we have 0 = 2 tan θ, which implies tan θ = 0, and thus θ = 0. Similarly, when x = 2, we have 2 = 2 tan θ, which implies tan θ = 1, and thus θ = π/4. These new limits of integration, 0 and π/4, are crucial for evaluating the definite integral in the θ domain. With the substitution, differential, and new limits in hand, we're ready to rewrite the integral entirely in terms of θ. This transformation is the heart of the trigonometric substitution technique, allowing us to leverage trigonometric identities and simplify the expression. The process may seem intricate, but each step is logical and deliberate, guided by the principles of calculus. The careful execution of these steps is what transforms a challenging integral into a solvable one.

Transforming the Integral: From x to θ

With the substitution x = 2 tan θ, dx = 2 sec² θ dθ, and the new limits of integration (0 and π/4) determined, we're poised to rewrite the integral ∫₀² dx/(x²+4) entirely in terms of θ. This transformation is the pivotal moment where the power of trigonometric substitution truly shines. Substituting x = 2 tan θ into the denominator, we get x² + 4 = (2 tan θ)² + 4 = 4 tan² θ + 4. Factoring out the 4, we have 4(tan² θ + 1). Now, recall the trigonometric identity 1 + tan² θ = sec² θ. Applying this identity, we can further simplify the denominator to 4 sec² θ. This simplification is a direct consequence of our strategic substitution, highlighting the elegance of the technique. Now, let's substitute dx = 2 sec² θ dθ into the numerator. Our integral now becomes:

∫₀^(π/4) (2 sec² θ dθ) / (4 sec² θ)

Observe the beautiful cancellation that occurs! The sec² θ terms in the numerator and denominator gracefully eliminate each other, leaving us with:

∫₀^(π/4) (2 dθ) / 4 = (1/2) ∫₀^(π/4) dθ

This simplified integral is significantly easier to handle than the original. The trigonometric substitution has effectively transformed a complex expression into a basic integral. This step-by-step transformation showcases the power of mathematical manipulation and the importance of choosing the right technique. The journey from the original integral to this simplified form underscores the beauty of calculus – its ability to break down seemingly insurmountable problems into manageable steps.

Evaluating the Simplified Integral: A Triumphant Conclusion

Having successfully transformed the integral into a simplified form, (1/2) ∫₀^(π/4) dθ, we're now on the verge of a triumphant conclusion. Evaluating this integral is a straightforward process, leveraging the fundamental theorem of calculus. The integral of dθ with respect to θ is simply θ. Therefore, we have:

(1/2) ∫₀^(π/4) dθ = (1/2) [θ]₀^(π/4)

Now, we substitute the limits of integration, π/4 and 0, into the expression:

(1/2) [π/4 - 0] = (1/2) * (π/4) = π/8

And there we have it! The definite integral ∫₀² dx/(x²+4) evaluates to π/8. This result is the culmination of our step-by-step journey, from recognizing the applicability of trigonometric substitution to the final evaluation. This problem serves as a powerful example of how strategic substitutions can simplify complex integrals. The elegance of the solution lies in the seamless integration of trigonometric identities and the fundamental theorem of calculus. It's not just about arriving at the answer; it's about appreciating the process and the underlying mathematical principles. The result, π/8, is not merely a number; it's a testament to the power of calculus and its ability to solve seemingly challenging problems.

In conclusion, our journey through the evaluation of the definite integral ∫₀² dx/(x²+4) has been a testament to the elegance and power of calculus. By employing the technique of trigonometric substitution, we transformed a seemingly complex integral into a manageable one, ultimately arriving at the solution π/8. This exploration has highlighted several key concepts: the importance of recognizing patterns in integrands, the strategic application of substitutions, and the fundamental role of trigonometric identities in simplification. But beyond the specific steps and techniques, this exercise has underscored a broader point: mathematics is not just about formulas and calculations; it's about problem-solving, logical reasoning, and the appreciation of beautiful solutions. The integral we tackled is a microcosm of the larger world of calculus, a world filled with challenges and opportunities for discovery. By mastering the techniques presented here, we equip ourselves to explore that world with confidence and curiosity. The journey of learning calculus is a continuous one, but each problem solved, each technique mastered, brings us closer to a deeper understanding of the mathematical universe. The result, π/8, stands as a symbol of our success, but the real reward lies in the knowledge and skills we've gained along the way. This knowledge empowers us to tackle future challenges and to appreciate the profound beauty of mathematics. The power of calculus is undeniable.