Integrating 1/(x^2 + 1/25) Step-by-Step Solution And Explanation

Jul 11, 2025 by ADMIN 65 views

Integrating 1/(x^2 + 1/25) A Comprehensive Guide

In the realm of calculus, integration stands as a fundamental operation, playing a pivotal role in solving a myriad of problems across various scientific and engineering disciplines. Among the diverse range of integrals encountered, those involving rational functions, particularly those with quadratic denominators, often pose a unique challenge. In this comprehensive guide, we will delve into the intricacies of integrating the specific function 1/(x^2 + 1/25), providing a step-by-step approach that will not only equip you with the necessary skills to tackle this particular integral but also enhance your understanding of integration techniques in general.

Understanding the Integral

At its core, the integral ∫ 1/(x^2 + 1/25) dx represents the area under the curve of the function f(x) = 1/(x^2 + 1/25). This seemingly simple expression conceals a fascinating interplay of mathematical concepts, inviting us to employ a variety of techniques to arrive at its solution. The key to unraveling this integral lies in recognizing its resemblance to a standard integral form, which we will explore in detail in the subsequent sections. This integral falls under the category of integrals involving quadratic expressions in the denominator. Mastering these types of integrals is crucial for anyone delving into calculus and its applications. The expression 1/(x^2 + 1/25) might initially seem daunting, but with the right approach, it can be elegantly solved. This process not only provides a solution but also reinforces fundamental integration principles. Understanding the structure of the integrand, 1/(x^2 + 1/25), is the first step. We observe that the denominator is a sum of squares, which hints at the possibility of using trigonometric substitution or a formula involving the arctangent function. The constant term 1/25 can be rewritten as (1/5)^2, making the expression even more suggestive of a form that can be integrated using a standard formula. This initial recognition is crucial in guiding our approach and selecting the appropriate technique. Recognizing this pattern is essential for choosing the correct integration strategy. Furthermore, understanding the properties of the arctangent function is crucial for successfully evaluating this integral. The arctangent function is the inverse of the tangent function, and it plays a significant role in integrals involving the form 1/(x^2 + a^2). Its derivative, 1/(1 + x^2), is a key component in the integration process. The process of integrating 1/(x^2 + 1/25) not only provides a specific solution but also enhances your problem-solving skills in calculus. By understanding the underlying principles and techniques, you can apply them to a wide range of similar problems. The integral ∫ 1/(x^2 + 1/25) dx is a classic example that showcases the power and elegance of calculus. The integration of this function demonstrates how seemingly complex problems can be solved using a systematic approach and a solid understanding of mathematical principles. It serves as a valuable exercise for students and practitioners alike, reinforcing the importance of mastering fundamental calculus concepts.

Recognizing the Standard Integral Form

The crucial step in solving this integral is recognizing its connection to the standard integral form: ∫ 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C, where 'a' is a constant and 'C' represents the constant of integration. This formula is a cornerstone in integral calculus, providing a direct route to integrating functions with a specific quadratic denominator. By recognizing this standard form, we can avoid more complex integration techniques, such as trigonometric substitution, and arrive at the solution efficiently. Comparing our integral, ∫ 1/(x^2 + 1/25) dx, with the standard form, we can identify that 'a^2' corresponds to 1/25. This allows us to deduce that 'a' is equal to 1/5. This identification is a critical step, as it allows us to directly apply the standard integral formula. Without recognizing this pattern, we might resort to more complicated methods, making the integration process significantly more challenging. The standard integral form ∫ 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C is derived using trigonometric substitution. However, memorizing this formula and recognizing its applicability can save considerable time and effort when solving integrals of this type. This formula is a powerful tool in the arsenal of any calculus student or practitioner. Understanding the derivation of this formula can provide deeper insights into the relationship between trigonometric functions and integration. However, for practical purposes, memorizing the formula and recognizing its applicability is often sufficient. Recognizing the standard form is not just about memorizing a formula; it's about developing pattern recognition skills. This ability to identify familiar patterns in mathematical expressions is crucial for problem-solving in calculus and other areas of mathematics. The more you practice recognizing these patterns, the more efficient and confident you will become in your problem-solving abilities. In our case, recognizing the standard form allows us to bypass more complex methods and directly apply the formula. This highlights the importance of developing a strong understanding of standard integral forms. The ability to recognize and apply standard integral forms is a hallmark of a proficient calculus student. It demonstrates a deep understanding of fundamental concepts and the ability to apply them effectively. This skill is invaluable for solving a wide range of integration problems.

Applying the Formula

Now that we have identified the standard integral form and determined the value of 'a' (1/5), we can directly apply the formula. Substituting a = 1/5 into the formula ∫ 1/(x^2 + a^2) dx = (1/a) arctan(x/a) + C, we get: ∫ 1/(x^2 + 1/25) dx = (1/(1/5)) arctan(x/(1/5)) + C. This substitution is a straightforward application of the formula, but it's important to perform it carefully to avoid errors. A common mistake is to misinterpret the value of 'a' or to incorrectly substitute it into the formula. Therefore, double-checking your work is always recommended. After substituting the value of 'a', we can simplify the expression. Dividing by a fraction is the same as multiplying by its reciprocal, so 1/(1/5) simplifies to 5. Similarly, x/(1/5) simplifies to 5x. This simplification leads us to the final form of the integral. The simplification process is crucial for obtaining the most concise and elegant form of the solution. While the initial expression obtained after substitution is technically correct, simplifying it makes the solution easier to understand and use. This step demonstrates the importance of not only finding the correct answer but also presenting it in the most clear and efficient manner. Performing the simplification, we obtain: ∫ 1/(x^2 + 1/25) dx = 5 arctan(5x) + C. This is the final result of our integration. This result demonstrates the power of recognizing standard integral forms and applying them correctly. It also highlights the importance of simplifying the expression after substitution to obtain the most concise and understandable answer. The constant of integration, 'C', is a crucial part of the solution. It represents the family of antiderivatives of the function 1/(x^2 + 1/25). Since the derivative of a constant is zero, any constant can be added to the antiderivative without changing its derivative. Therefore, we include 'C' to represent all possible antiderivatives. The inclusion of the constant of integration is a fundamental aspect of indefinite integration. It acknowledges the fact that there are infinitely many functions that could have the same derivative. By including 'C', we ensure that our solution represents the entire family of antiderivatives. The final result, 5 arctan(5x) + C, is a concise and elegant solution to the integral. It demonstrates the power of calculus and the importance of understanding fundamental integration techniques.

The Solution

Therefore, the integral of 1/(x^2 + 1/25) is 5 arctan(5x) + C, where C is the constant of integration. This is our final answer. This solution encapsulates the entire process we have undertaken, from recognizing the standard integral form to applying the formula and simplifying the result. It serves as a testament to the power of calculus and the elegance of mathematical solutions. The solution 5 arctan(5x) + C is not just a numerical answer; it represents a function. This function, when differentiated, will yield the original integrand, 1/(x^2 + 1/25). This is the fundamental concept of integration – finding the antiderivative of a function. The arctangent function plays a crucial role in this solution. It arises naturally from the integral of functions with a quadratic denominator. Understanding the properties of the arctangent function is essential for mastering this type of integration. The constant of integration, C, is a reminder that there are infinitely many solutions to an indefinite integral. Each value of C corresponds to a different antiderivative, all of which differ by a constant. This concept is fundamental to understanding indefinite integration. The solution 5 arctan(5x) + C can be verified by differentiating it. The derivative of 5 arctan(5x) is 5 * (1/(1 + (5x)^2)) * 5 = 25/(1 + 25x^2) = 1/(1/25 + x^2), which is the original integrand. This verification process confirms the correctness of our solution. This process of verifying the solution by differentiation is a valuable technique for checking your work in integration. It ensures that you have indeed found an antiderivative of the original function. The integral ∫ 1/(x^2 + 1/25) dx = 5 arctan(5x) + C serves as a valuable example for understanding integration techniques. It demonstrates the importance of recognizing standard forms, applying formulas correctly, and simplifying the result. This example can be used as a template for solving other integrals of similar form. Mastering this type of integral is a crucial step in developing your calculus skills. It provides a foundation for tackling more complex integration problems. The solution 5 arctan(5x) + C is a testament to the beauty and power of mathematics. It demonstrates how seemingly complex problems can be solved using a systematic approach and a solid understanding of fundamental principles.

Conclusion

In conclusion, integrating 1/(x^2 + 1/25) involves recognizing the standard integral form, applying the corresponding formula, and simplifying the result. The final answer is 5 arctan(5x) + C. This exercise not only provides a specific solution but also reinforces essential integration techniques applicable to a broader range of problems in calculus. Mastering these techniques is crucial for success in calculus and its applications. This specific example serves as a stepping stone to understanding more complex integration scenarios. The process of recognizing patterns, applying formulas, and simplifying expressions is a fundamental skill in mathematics. This skill is not limited to calculus; it is applicable to various other areas of mathematics and science. The integral ∫ 1/(x^2 + 1/25) dx = 5 arctan(5x) + C is a classic example that demonstrates the elegance and power of calculus. It highlights the importance of understanding fundamental concepts and applying them effectively. This type of integral is frequently encountered in various applications of calculus, such as physics, engineering, and economics. Therefore, mastering its solution is a valuable skill. The process of integrating this function reinforces the connection between differentiation and integration. Integration is the reverse process of differentiation, and understanding this relationship is crucial for success in calculus. The solution 5 arctan(5x) + C provides a concrete example of how these two fundamental operations are related. The constant of integration, C, reminds us of the nature of indefinite integration. It highlights the fact that there are infinitely many antiderivatives for a given function, all differing by a constant. The ability to solve integrals like this one is a valuable asset in various fields. It allows us to model and solve real-world problems involving rates of change and accumulation. The study of calculus is essential for anyone pursuing a career in science, engineering, or mathematics. This example demonstrates the practical relevance of calculus and its ability to solve complex problems. The solution 5 arctan(5x) + C is a concise and elegant result that encapsulates the beauty of mathematical solutions. It serves as a testament to the power of human intellect and the ability to unravel complex problems through logical reasoning and mathematical techniques.