Solving ∫ 1/(cos⁴(3x+2) + Cos²(3x+2)sin²(3x+2)) Dx A Step-by-Step Guide

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Introduction

In the realm of calculus, integrals often present intricate challenges, demanding a blend of trigonometric identities, algebraic manipulations, and strategic substitutions. This article embarks on a comprehensive exploration of the integral ∫ 1/(cos⁴(3x+2) + cos²(3x+2)sin²(3x+2)) dx, dissecting the problem step-by-step to reveal the underlying techniques and arrive at the solution. We will delve into the nuances of trigonometric functions, their interrelationships, and how these can be leveraged to simplify complex expressions. Through a meticulous process of transformation and integration, we aim to provide a clear understanding of the problem-solving approach, making it accessible to students and enthusiasts alike. This exploration will not only solve the integral but also enhance your skills in manipulating trigonometric integrals, a crucial aspect of advanced calculus. Our approach will emphasize clarity and logical progression, ensuring that each step is well-explained and justified. From the initial simplification to the final integration, every detail will be carefully examined, offering valuable insights into the art of integral calculus. The journey through this integral will highlight the beauty and power of mathematical techniques, demonstrating how seemingly complex problems can be elegantly solved with the right tools and understanding. So, let's embark on this mathematical adventure and unravel the mysteries of this trigonometric integral together.

Simplifying the Integrand

To effectively tackle the integral ∫ 1/(cos⁴(3x+2) + cos²(3x+2)sin²(3x+2)) dx, our initial focus must be on simplifying the integrand. The denominator, cos⁴(3x+2) + cos²(3x+2)sin²(3x+2), appears complex, but we can streamline it using fundamental trigonometric identities. The key lies in recognizing the common factor of cos²(3x+2) within the denominator. By factoring out this term, we transform the denominator into a more manageable form. This initial step is crucial as it sets the stage for further simplification and eventual integration. Factoring not only reduces the complexity but also unveils a structure that is more amenable to manipulation. The process of simplification is often the cornerstone of solving complex integrals, and this case is no exception. The factored form allows us to apply trigonometric identities more effectively, paving the way for a solution. This stage is about strategic manipulation, using our knowledge of trigonometric relationships to make the problem more accessible. The simplified form of the integrand will then allow us to employ substitution or other integration techniques with greater ease. This initial transformation is not just about making the expression look simpler; it's about fundamentally changing its nature to make integration feasible. Let's delve into the specifics of this simplification process, showing each step in detail to ensure clarity and understanding.

Detailed Simplification Steps:

  1. Identify the common factor: In the denominator cos⁴(3x+2) + cos²(3x+2)sin²(3x+2), the common factor is cos²(3x+2).
  2. Factor out the common factor: Factoring cos²(3x+2) from the denominator gives us cos²(3x+2) [cos²(3x+2) + sin²(3x+2)].
  3. Apply the Pythagorean identity: Recall the fundamental trigonometric identity: sin²(θ) + cos²(θ) = 1. In our case, θ = (3x+2), so cos²(3x+2) + sin²(3x+2) = 1.
  4. Simplify the denominator: Substituting the identity, the denominator simplifies to cos²(3x+2) * 1 = cos²(3x+2).
  5. Rewrite the integrand: The integral now transforms from ∫ 1/(cos⁴(3x+2) + cos²(3x+2)sin²(3x+2)) dx to ∫ 1/cos²(3x+2) dx.

This simplification is a pivotal step, transforming a seemingly daunting integral into a more recognizable form. The application of the Pythagorean identity is a classic technique in trigonometric integration, and here it proves to be particularly effective. The simplified integrand, 1/cos²(3x+2), is a standard trigonometric function that we can easily integrate. This transformation underscores the importance of recognizing and utilizing fundamental trigonometric identities to navigate complex integrals. By reducing the complexity of the integrand, we have set the stage for a straightforward integration process, making the problem significantly more manageable.

Transforming to a Standard Integral

With the integrand simplified to 1/cos²(3x+2), we're now poised to transform it into a standard integral form. This step involves recognizing the relationship between 1/cos²(θ) and a well-known trigonometric function. The reciprocal of cosine is secant, thus 1/cos²(3x+2) is equivalent to sec²(3x+2). This transformation is a critical link in our solution, connecting the simplified form to a directly integrable function. Recognizing this equivalence allows us to leverage our knowledge of standard integrals. The function sec²(θ) is a fundamental trigonometric derivative, making it a prime candidate for direct integration. The process of transforming the integrand into a standard form is a common strategy in integral calculus. It involves identifying familiar patterns and relationships that allow us to express the integral in terms of functions whose antiderivatives are known. In this case, the transformation from 1/cos²(3x+2) to sec²(3x+2) is a direct application of trigonometric definitions, but its impact on the integrability of the expression is profound. This step highlights the importance of mastering trigonometric identities and reciprocal relationships. A solid understanding of these fundamentals is essential for navigating the complexities of integral calculus. The transformation to a standard form not only simplifies the integration process but also provides a clear path to the final solution. By recognizing and applying this transformation, we are one step closer to solving the integral.

Detailed Transformation Steps:

  1. Recognize the reciprocal relationship: Recall that sec(θ) = 1/cos(θ). Therefore, sec²(θ) = 1/cos²(θ).
  2. Apply the relationship to the integrand: Substitute 1/cos²(3x+2) with sec²(3x+2). The integral now becomes ∫ sec²(3x+2) dx.

This transformation is a direct application of a fundamental trigonometric definition, but it's crucial for progressing towards the solution. The function sec²(3x+2) is a standard trigonometric function whose integral we know. This step illustrates the power of recognizing trigonometric relationships and using them to simplify integrals. The transformed integral, ∫ sec²(3x+2) dx, is now in a form that we can easily integrate using standard techniques. This transformation underscores the importance of having a strong foundation in trigonometric identities and reciprocal relationships. By converting the integrand into a standard form, we have made the integration process significantly more straightforward.

Integration Using Substitution

Now that we have transformed the integral into the standard form ∫ sec²(3x+2) dx, we proceed with integration. The presence of the term (3x+2) within the secant function suggests a u-substitution. This technique is a powerful tool in integral calculus, allowing us to simplify complex integrals by introducing a new variable. The key to a successful u-substitution is identifying a suitable expression within the integrand whose derivative also appears, or can be made to appear, in the integral. In this case, letting u = 3x+2 is a strategic choice. The derivative of u with respect to x is du/dx = 3, which is a constant that can be easily accommodated within the integral. This substitution will transform the integral into a simpler form involving sec²(u), which has a known antiderivative. The u-substitution method is not just about changing variables; it's about simplifying the structure of the integral to make it more amenable to integration. By carefully selecting the substitution, we can often reduce a complex integral to a basic form that we can easily handle. The process of u-substitution involves several steps, including defining the new variable, finding its derivative, and substituting both the variable and its differential into the integral. The goal is to express the entire integral in terms of the new variable, allowing us to apply standard integration rules. This technique is a cornerstone of integral calculus, and mastering it is essential for solving a wide range of integrals. Let's delve into the details of this substitution and see how it simplifies our integral.

Detailed Substitution Steps:

  1. Define the substitution: Let u = 3x+2.
  2. Find the derivative: Differentiating both sides with respect to x, we get du/dx = 3.
  3. Solve for dx: Rearranging the equation, we have dx = du/3.
  4. Substitute into the integral: Replace (3x+2) with u and dx with du/3 in the integral ∫ sec²(3x+2) dx. This gives us ∫ sec²(u) (du/3).
  5. Simplify the integral: The constant 1/3 can be factored out of the integral, resulting in (1/3) ∫ sec²(u) du.

This u-substitution has effectively simplified the integral, transforming it into a basic form that we can easily integrate. The constant factor of 1/3 is a result of the chain rule in differentiation and must be carefully accounted for during the integration process. The integral (1/3) ∫ sec²(u) du is now a standard integral that we can directly evaluate. This transformation underscores the power of u-substitution in simplifying integrals and making them more manageable. By introducing a new variable, we have reduced the complexity of the integrand and made the integration process straightforward.

Evaluating the Integral

Having simplified the integral to (1/3) ∫ sec²(u) du, we are now in a position to evaluate it directly. The integral of sec²(u) is a fundamental result in calculus, namely tan(u). This antiderivative is a cornerstone of trigonometric integration and should be readily recognized. The integral ∫ sec²(u) du = tan(u) + C, where C is the constant of integration. This constant is a crucial element of indefinite integrals, representing the family of functions that have the same derivative. It's important to include this constant in our final answer to reflect the general solution of the integral. Once we have found the antiderivative in terms of u, the final step is to substitute back for u in terms of x. This substitution reverses the initial u-substitution and expresses the result in terms of the original variable. The process of evaluating the integral involves recognizing the antiderivative of the simplified integrand and then accounting for the constant of integration. This step is a culmination of all the previous simplification and transformation efforts, bringing us to the solution of the integral. The ability to recognize standard antiderivatives is a key skill in integral calculus, and the integral of sec²(u) is a prime example of a fundamental result that should be memorized. Let's now complete the evaluation and express the final result in terms of x.

Detailed Evaluation Steps:

  1. Find the antiderivative: The integral of sec²(u) is tan(u). Therefore, ∫ sec²(u) du = tan(u) + C.
  2. Substitute back for u: Replace u with (3x+2) in the antiderivative. This gives us tan(3x+2) + C.
  3. Include the constant factor: Multiply the result by the constant factor (1/3) from the u-substitution. This gives us (1/3)tan(3x+2) + C.

This evaluation completes the integration process, providing us with the antiderivative of the original integrand. The final result, (1/3)tan(3x+2) + C, represents the general solution to the integral ∫ 1/(cos⁴(3x+2) + cos²(3x+2)sin²(3x+2)) dx. The constant of integration, C, is included to represent the family of functions that have the same derivative. This evaluation underscores the importance of knowing standard antiderivatives and carefully accounting for constants of integration. By combining the u-substitution technique with the fundamental antiderivative of sec²(u), we have successfully solved the integral.

Conclusion

In conclusion, the integral ∫ 1/(cos⁴(3x+2) + cos²(3x+2)sin²(3x+2)) dx can be elegantly solved through a series of strategic simplifications, transformations, and integration techniques. The initial step involved simplifying the integrand by factoring out cos²(3x+2) from the denominator and applying the Pythagorean trigonometric identity. This simplification paved the way for a crucial transformation, where we recognized that 1/cos²(3x+2) is equivalent to sec²(3x+2). This transformation allowed us to express the integral in a standard form, ∫ sec²(3x+2) dx, which is amenable to integration. To handle the term (3x+2) within the secant function, we employed the u-substitution technique, letting u = 3x+2. This substitution transformed the integral into a simpler form, (1/3) ∫ sec²(u) du. The integral of sec²(u) is a fundamental result in calculus, namely tan(u). By integrating and then substituting back for u in terms of x, we arrived at the final solution: (1/3)tan(3x+2) + C, where C is the constant of integration. This journey through the integral highlights the power of combining trigonometric identities, algebraic manipulations, and integration techniques. The solution underscores the importance of recognizing standard antiderivatives and carefully accounting for constants of integration. The process demonstrates how seemingly complex integrals can be tackled effectively by breaking them down into manageable steps. The u-substitution technique, in particular, proved to be a valuable tool in simplifying the integral and making it solvable. The final result, (1/3)tan(3x+2) + C, represents the general solution to the integral and completes our exploration. This exercise not only solves a specific integral but also reinforces the fundamental principles of integral calculus and their application in problem-solving.