Evaluating The Integral Of Sin(6x)sec⁵(6x) Dx A Step-by-Step Guide
Evaluating integrals involving trigonometric functions often requires a blend of trigonometric identities, substitution techniques, and a keen eye for simplification. This article delves into the step-by-step evaluation of the integral ∫sin(6x)sec⁵(6x) dx, emphasizing the underlying concepts and methods employed. We will navigate through the process, highlighting crucial steps and providing explanations to ensure a clear understanding of the solution. The goal is not just to arrive at the final answer but to equip you with the knowledge to tackle similar integration problems with confidence. This journey through trigonometric integration will showcase how seemingly complex integrals can be systematically solved by leveraging fundamental principles and techniques. Trigonometric integrals form a cornerstone of calculus, appearing in various scientific and engineering applications, making their mastery essential for anyone pursuing these fields. Whether you're a student grappling with calculus or a professional seeking a refresher, this comprehensive guide will illuminate the path to solving integrals of this nature. As we embark on this mathematical exploration, remember that each step is a building block, contributing to the final solution and enhancing your problem-solving skills. So, let's dive into the fascinating world of trigonometric integrals and unravel the intricacies of ∫sin(6x)sec⁵(6x) dx.
1. Initial Setup and Trigonometric Identities
To begin, we're presented with the integral: ∫sin(6x)sec⁵(6x) dx. The first step in evaluating this integral involves recognizing the trigonometric functions and their relationships. Specifically, we know that sec(x) is the reciprocal of cos(x), meaning sec(x) = 1/cos(x). Thus, sec⁵(6x) can be rewritten as 1/cos⁵(6x). This transformation allows us to express the integral in terms of sine and cosine functions, which are often easier to manipulate. By rewriting the integral, we are setting the stage for a substitution method, which is a powerful technique for simplifying integrals. This initial transformation is crucial because it bridges the gap between the given form and a more manageable one. Furthermore, it demonstrates the importance of recognizing fundamental trigonometric identities and their applicability in simplifying complex expressions. Without this initial step, the integral might appear daunting, but with this transformation, we pave the way for a more straightforward solution. This process underscores the idea that mastery of trigonometric identities is paramount in calculus, especially when dealing with integrals involving trigonometric functions. So, by converting sec⁵(6x) to 1/cos⁵(6x), we not only simplify the expression but also unlock the potential for further steps in the integration process. The initial setup is the foundation upon which the rest of the solution will be built, making it an indispensable part of the problem-solving strategy.
2. Rewriting the Integral
Following the initial setup, we rewrite the integral using the identity sec(x) = 1/cos(x). This gives us: ∫sin(6x) * (1/cos⁵(6x)) dx, which can be further simplified to ∫sin(6x)/cos⁵(6x) dx. Now, we can separate one cosine term from the denominator to pair it with the sine term. This manipulation yields ∫(sin(6x)/cos(6x)) * (1/cos⁴(6x)) dx. Recognizing that sin(x)/cos(x) is equivalent to tan(x), and 1/cos(x) is sec(x), we can rewrite the integral as ∫tan(6x)sec⁴(6x) dx. This transformation is a key step because it brings us closer to a form where a suitable substitution can be applied. The integral now involves tan(6x) and sec⁴(6x), which are related through the derivative of the tangent function. This form is significantly easier to work with compared to the original integral, showcasing the power of strategic rewriting in simplifying complex integrals. The manipulation highlights the importance of recognizing trigonometric relationships and how they can be leveraged to transform integrals into more manageable forms. By expressing the integral in terms of tangent and secant functions, we set the stage for the next step, which involves a strategic substitution that will further simplify the integral and allow us to find its antiderivative. This process demonstrates how a thoughtful approach to rewriting integrals can significantly ease the burden of integration.
3. Strategic Substitution
The strategic substitution is a cornerstone technique in calculus for simplifying integrals. In this case, observing the integral ∫tan(6x)sec⁴(6x) dx, we notice a potential substitution that can drastically simplify the expression. We can rewrite sec⁴(6x) as sec²(6x) * sec²(6x). Now, using the trigonometric identity sec²(x) = 1 + tan²(x), we can express one of the sec²(6x) terms as 1 + tan²(6x). This gives us ∫tan(6x) * (1 + tan²(6x)) * sec²(6x) dx. The appearance of tan(6x) and sec²(6x) suggests a u-substitution, where we let u = tan(6x). The derivative of u with respect to x, du/dx, is 6sec²(6x), which implies du = 6sec²(6x) dx. Consequently, sec²(6x) dx can be expressed as du/6. This substitution transforms the integral into a simpler form in terms of u, making it much easier to integrate. The beauty of u-substitution lies in its ability to reduce complex integrals to basic forms by strategically choosing a part of the integrand and its derivative. In this scenario, the choice of u = tan(6x) is driven by the presence of its derivative, sec²(6x), in the integral. By carefully selecting the substitution, we pave the way for a straightforward integration process. This step underscores the importance of recognizing patterns and relationships within the integrand to identify suitable substitutions that can lead to a simplified integral.
4. Applying the u-Substitution
Having identified the strategic substitution u = tan(6x), we now proceed to apply it to the integral. Recall that du = 6sec²(6x) dx, which means sec²(6x) dx = du/6. Substituting u and du into the integral ∫tan(6x)(1 + tan²(6x))sec²(6x) dx, we obtain ∫u(1 + u²)(du/6). This transformation effectively replaces the trigonometric functions with algebraic terms, making the integral much simpler to handle. The constant 1/6 can be factored out of the integral, resulting in (1/6)∫u(1 + u²) du. Next, we distribute u across the terms inside the parentheses, yielding (1/6)∫(u + u³) du. This form is now readily integrable using the power rule for integration. The power rule states that ∫xⁿ dx = (xⁿ⁺¹)/(n + 1) + C, where C is the constant of integration. By applying the u-substitution, we have successfully converted a complex trigonometric integral into a basic polynomial integral. This step demonstrates the effectiveness of u-substitution in simplifying integrals by changing the variable of integration. The new integral is now in a form that can be easily solved using standard integration techniques, highlighting the power of strategic substitutions in calculus. The transition from a trigonometric integral to a polynomial integral is a testament to the versatility and utility of the u-substitution method.
5. Integrating with Respect to u
With the u-substitution applied, we are now faced with the integral (1/6)∫(u + u³) du. Integrating this expression with respect to u is a straightforward application of the power rule. The power rule of integration states that ∫xⁿ dx = (xⁿ⁺¹)/(n + 1) + C, where C is the constant of integration. Applying this rule to our integral, we get: (1/6) * [∫u du + ∫u³ du]. The integral of u with respect to u is u²/2, and the integral of u³ with respect to u is u⁴/4. Therefore, the integral becomes (1/6) * [u²/2 + u⁴/4] + C, where C is the constant of integration. This step highlights the simplicity of integrating polynomial functions compared to trigonometric functions. The u-substitution has effectively transformed a complex trigonometric integral into a basic polynomial integral, making the integration process much easier. The result is a polynomial expression in terms of u, which is a direct consequence of the strategic substitution we employed earlier. By integrating with respect to u, we have found the antiderivative in terms of the new variable. However, to obtain the final answer, we need to substitute back for u in terms of the original variable, x. This process of integrating with respect to the new variable and then substituting back is a hallmark of the u-substitution technique.
6. Substituting Back for x
Having integrated with respect to u, we now have the expression (1/6) * [u²/2 + u⁴/4] + C. To express the result in terms of the original variable, x, we need to substitute back u = tan(6x). This gives us (1/6) * [(tan²(6x))/2 + (tan⁴(6x))/4] + C. We can simplify this expression further by distributing the 1/6 across the terms, resulting in (tan²(6x))/12 + (tan⁴(6x))/24 + C. This is the antiderivative of the original integral, expressed in terms of x. The constant of integration, C, is crucial because it represents the family of functions that have the same derivative. The act of substituting back for x completes the integration process, providing us with the solution to the original integral. This step emphasizes the importance of remembering to revert back to the original variable after performing a substitution. The final expression is a combination of tangent functions, which is a direct result of the u-substitution and the trigonometric identities used throughout the process. By substituting back for x, we have successfully transformed the antiderivative from u-space to x-space, providing us with the solution in the context of the original problem. This process underscores the importance of the substitution method in simplifying integrals and the necessity of reverting back to the original variable to obtain the final answer.
7. Final Result and Conclusion
After substituting back for x and simplifying, we arrive at the final result: (tan²(6x))/12 + (tan⁴(6x))/24 + C. This is the evaluated integral of ∫sin(6x)sec⁵(6x) dx. The constant of integration, C, is a crucial component of the final answer, reminding us that there are infinitely many functions that have the same derivative. In conclusion, evaluating this integral involved a series of strategic steps, including the use of trigonometric identities, rewriting the integral, applying a u-substitution, and integrating with respect to the new variable. Each step was carefully chosen to simplify the integral and make it more manageable. The process highlights the importance of recognizing trigonometric relationships, mastering substitution techniques, and applying the power rule of integration. Furthermore, it demonstrates how a complex integral can be systematically solved by breaking it down into smaller, more manageable parts. This journey through trigonometric integration has not only provided us with the solution to this specific problem but has also reinforced the fundamental principles and techniques that are essential for tackling a wide range of integration problems. The final result is a testament to the power of calculus and the elegance of mathematical problem-solving. By understanding and applying these techniques, we can confidently approach and solve a variety of integrals, expanding our mathematical toolkit and enhancing our problem-solving skills.
This article walked through a detailed, step-by-step solution to the integral ∫sin(6x)sec⁵(6x) dx, demonstrating the power of strategic simplification and substitution in calculus.
