Evaluating The Integral Of Cos(5x)sin(3x) A Step-by-Step Guide
Introduction
In this article, we will delve into the step-by-step evaluation of the integral ∫ cos(5x) sin(3x) dx. This integral falls under the category of trigonometric integrals, which often require the application of trigonometric identities to simplify the integrand before direct integration can be performed. Mastering these techniques is crucial for anyone studying calculus, physics, or engineering, where trigonometric integrals frequently appear in various problem-solving scenarios. Our approach will utilize the product-to-sum trigonometric identity, a powerful tool that transforms the product of trigonometric functions into a sum, making the integration process significantly easier. We will explore how this identity allows us to break down a complex integral into simpler, more manageable parts. By following this detailed explanation, you will gain a solid understanding of how to tackle similar trigonometric integrals effectively.
Trigonometric integrals are an essential topic in calculus and often appear in various fields such as physics and engineering. The integral ∫ cos(5x) sin(3x) dx is a classic example that requires the use of trigonometric identities to simplify the integrand before integration. Direct integration might seem challenging at first, but by applying the appropriate trigonometric identity, we can transform the product of cosine and sine into a sum, making the integral much easier to solve. This article will provide a step-by-step explanation of how to evaluate this integral, ensuring a clear and thorough understanding of the process. Understanding trigonometric integrals is crucial for anyone studying calculus, physics, or engineering, where these integrals frequently appear in problem-solving. The method we will use involves transforming the product of trigonometric functions into a sum using a trigonometric identity, a common technique in calculus. This transformation simplifies the integral, allowing us to integrate each term separately. The step-by-step approach will provide clarity and help in understanding the underlying principles, ensuring you can apply this method to similar problems. Moreover, we will discuss the importance of understanding and applying trigonometric identities in calculus. These identities are not only useful for solving integrals but also for simplifying trigonometric expressions and solving trigonometric equations. The ability to manipulate trigonometric functions is a valuable skill in many areas of mathematics and its applications. The integral ∫ cos(5x) sin(3x) dx serves as an excellent example for learning these techniques. By the end of this article, you will have a comprehensive understanding of how to evaluate this integral and be well-prepared to tackle similar problems involving trigonometric functions. This skill is fundamental for further studies in calculus and its applications in various scientific and engineering disciplines. The step-by-step methodology will also help in developing problem-solving strategies, which are crucial for success in advanced mathematics courses.
Step 1: Apply the Product-to-Sum Identity
To begin evaluating the integral ∫ cos(5x) sin(3x) dx, the first crucial step involves applying the appropriate trigonometric identity. In this case, we leverage the product-to-sum identity, which is specifically designed to handle integrals involving the product of sine and cosine functions. This identity transforms the product into a sum or difference of trigonometric functions, making the integration process significantly simpler. The product-to-sum identity we will use is: sin(A) cos(B) = (1/2)[sin(A + B) + sin(A - B)]. This identity is a cornerstone for solving many trigonometric integrals, and understanding its application is essential. By correctly applying this identity, we break down the complex integral into more manageable parts that can be integrated separately. The transformation allows us to avoid dealing with the product directly, which is often more challenging to integrate. This approach is a common strategy in calculus for handling integrals involving products of trigonometric functions, and it demonstrates the power of trigonometric identities in simplifying mathematical problems.
The product-to-sum identity is a powerful tool in calculus, especially when dealing with integrals involving products of trigonometric functions. For the integral ∫ cos(5x) sin(3x) dx, the relevant identity is sin(A) cos(B) = (1/2)[sin(A + B) + sin(A - B)]. This identity allows us to transform the product of sine and cosine into a sum of sines, which are much easier to integrate. Applying this identity is the key to simplifying the integral and making it solvable. Without this transformation, the integral would be significantly more complex to handle. The identity is derived from the sum and difference formulas for sine and cosine, and it's a fundamental tool in trigonometric manipulation. By substituting A = 3x and B = 5x into the identity, we can rewrite the integrand, making the integration process straightforward. This technique is not only useful for this particular integral but also for a wide range of trigonometric integrals. Understanding and applying such identities is a crucial skill for anyone studying calculus and related fields. The application of this identity is a prime example of how trigonometric identities can simplify complex mathematical problems. It allows us to break down a seemingly difficult integral into simpler, more manageable parts. The result of this transformation is a sum of sine functions, each of which can be integrated using basic calculus techniques. The importance of this step cannot be overstated, as it sets the stage for the rest of the solution. Without this simplification, the integral would remain a daunting challenge. By mastering the use of product-to-sum identities, you will be well-equipped to tackle a variety of trigonometric integrals and other problems in calculus.
Step 2: Substitute and Simplify
Having recognized the need to apply the product-to-sum identity, the next crucial step is to substitute the specific values from our integral, ∫ cos(5x) sin(3x) dx, into the identity sin(A) cos(B) = (1/2)[sin(A + B) + sin(A - B)]. Here, we identify A = 3x and B = 5x. Plugging these values into the identity gives us: cos(5x) sin(3x) = (1/2)[sin(3x + 5x) + sin(3x - 5x)]. This substitution is a direct application of the identity and is fundamental in transforming the integral into a simpler form. The next part of this step involves simplifying the expression. We combine the terms inside the sine functions to obtain: cos(5x) sin(3x) = (1/2)[sin(8x) + sin(-2x)]. This simplification makes the integral much more manageable, as we now have a sum of sine functions rather than a product of sine and cosine. Recognizing that sin(-x) = -sin(x), we can further simplify the expression to: cos(5x) sin(3x) = (1/2)[sin(8x) - sin(2x)]. This final simplified form is the key to easily integrating the expression. By breaking down the complex product into a sum of simpler sine functions, we set the stage for the final integration step. This meticulous substitution and simplification process highlights the importance of algebraic manipulation in calculus, particularly when dealing with trigonometric functions.
The substitution and simplification step is crucial in evaluating the integral ∫ cos(5x) sin(3x) dx. After identifying the product-to-sum identity sin(A) cos(B) = (1/2)[sin(A + B) + sin(A - B)], we substitute A = 3x and B = 5x into the identity. This gives us cos(5x) sin(3x) = (1/2)[sin(3x + 5x) + sin(3x - 5x)]. This substitution is the heart of transforming the original integral into a more manageable form. The next part of this step is to simplify the expression by combining the terms inside the sine functions. This leads to cos(5x) sin(3x) = (1/2)[sin(8x) + sin(-2x)]. Simplifying further requires recognizing the property of sine that sin(-x) = -sin(x). Applying this property, we can rewrite the expression as cos(5x) sin(3x) = (1/2)[sin(8x) - sin(2x)]. This simplified form is much easier to integrate because it separates the product into a sum of sine functions. Each sine function can be integrated directly using standard calculus techniques. The simplification process is a critical skill in calculus, and it often involves applying trigonometric identities and algebraic manipulations to transform complex expressions into simpler ones. This step highlights the importance of understanding the properties of trigonometric functions and how to use them effectively. By carefully substituting and simplifying, we have converted the original integral into a form that is readily integrable. This approach is a common strategy in calculus for dealing with integrals involving trigonometric functions. The meticulous attention to detail in this step ensures that the final integration will be straightforward and accurate.
Step 3: Integrate the Simplified Expression
With the expression now simplified to (1/2)[sin(8x) - sin(2x)], we can proceed to integrate each term separately. This step involves applying the fundamental rules of integration to sine functions. The integral of sin(ax) is -(1/a)cos(ax) + C, where C is the constant of integration. This rule is a direct result of the derivative of cosine being negative sine. Applying this rule to our expression, we integrate (1/2)sin(8x) and -(1/2)sin(2x) separately. The integral of (1/2)sin(8x) is (1/2) * (-(1/8)cos(8x)) = -(1/16)cos(8x). Similarly, the integral of -(1/2)sin(2x) is -(1/2) * (-(1/2)cos(2x)) = (1/4)cos(2x). Therefore, the integral of the entire expression is the sum of these two integrals. This step is crucial as it transforms the simplified trigonometric expression into its antiderivative, which represents the solution to the integral. Understanding the rules of integration for trigonometric functions is essential for this step. The constant of integration, C, is added to the final result because the derivative of a constant is zero, meaning there are infinitely many possible constant terms in the antiderivative. This integration step showcases the core principles of calculus and how they apply to trigonometric functions.
Integrating the simplified expression is a crucial step in evaluating the integral ∫ cos(5x) sin(3x) dx. After simplifying the integrand to (1/2)[sin(8x) - sin(2x)], we can now integrate each term separately. This involves applying the basic integration rule for sine functions: ∫ sin(ax) dx = -(1/a)cos(ax) + C, where C is the constant of integration. This rule is fundamental in calculus and is derived from the derivative of cosine. Applying this rule to the first term, (1/2)sin(8x), we find its integral to be (1/2) * (-(1/8)cos(8x)) = -(1/16)cos(8x). For the second term, -(1/2)sin(2x), its integral is -(1/2) * (-(1/2)cos(2x)) = (1/4)cos(2x). Now, we combine these two integrals to find the integral of the entire expression: ∫ (1/2)[sin(8x) - sin(2x)] dx = -(1/16)cos(8x) + (1/4)cos(2x) + C. The constant of integration, C, is added because the derivative of any constant is zero, so there are infinitely many possible constants in the antiderivative. This integration step is a direct application of calculus principles and demonstrates how to find the antiderivative of a trigonometric expression. The ability to integrate trigonometric functions is an essential skill in calculus and is used extensively in various applications. This step highlights the importance of understanding and applying the rules of integration correctly. The result is the general antiderivative of the original integrand, representing a family of functions that differ only by a constant. This understanding is crucial for solving a wide range of problems involving integrals and differential equations.
Step 4: Write the Final Solution
After integrating the simplified expression, the final step is to write out the complete solution, including the constant of integration. From the previous step, we found the integral of (1/2)[sin(8x) - sin(2x)] to be -(1/16)cos(8x) + (1/4)cos(2x). To complete the solution, we add the constant of integration, C, which accounts for the fact that the derivative of a constant is zero, and thus, there are infinitely many possible constant terms in the antiderivative. Therefore, the final solution to the integral ∫ cos(5x) sin(3x) dx is: -(1/16)cos(8x) + (1/4)cos(2x) + C. This is the antiderivative of the original integrand and represents the family of functions whose derivative is cos(5x) sin(3x). This final step is crucial as it provides the complete and accurate answer to the problem. Understanding the importance of the constant of integration is essential in calculus, as it ensures the generality of the solution. This result showcases the culmination of the steps taken, from applying trigonometric identities to performing the integration, and it provides a clear and concise answer to the integral problem.
The final step in evaluating the integral ∫ cos(5x) sin(3x) dx is to present the complete solution. After performing the integration in the previous step, we obtained -(1/16)cos(8x) + (1/4)cos(2x). To finalize the solution, we must include the constant of integration, C. This constant is essential because the derivative of any constant is zero, meaning that there are infinitely many possible constant terms in the antiderivative. Therefore, the complete solution to the integral is ∫ cos(5x) sin(3x) dx = -(1/16)cos(8x) + (1/4)cos(2x) + C. This final answer represents the family of functions whose derivative is equal to the original integrand, cos(5x) sin(3x). Writing the final solution correctly is crucial for completing the problem accurately. The constant of integration ensures that we have captured all possible antiderivatives. This step highlights the importance of attention to detail and the need to include all necessary components in the final answer. The solution we have obtained is a general solution, meaning it represents all possible antiderivatives of the given function. This understanding is fundamental in calculus and is essential for solving various types of integral problems. The final solution provides a clear and concise answer to the integral problem, demonstrating the effectiveness of the techniques and steps we have employed. This result showcases the culmination of the entire process, from applying trigonometric identities to performing the integration, and provides a solid foundation for tackling similar problems in the future.
Conclusion
In conclusion, we have successfully evaluated the integral ∫ cos(5x) sin(3x) dx by systematically applying trigonometric identities and integration techniques. The key to solving this integral was the use of the product-to-sum identity, which transformed the product of cosine and sine functions into a sum of sine functions, making the integration process significantly easier. We began by applying the identity sin(A) cos(B) = (1/2)[sin(A + B) + sin(A - B)], substituting A = 3x and B = 5x. This substitution allowed us to rewrite the integrand as (1/2)[sin(8x) - sin(2x)]. Next, we integrated each term separately, using the rule ∫ sin(ax) dx = -(1/a)cos(ax) + C. This yielded the result -(1/16)cos(8x) + (1/4)cos(2x). Finally, we added the constant of integration, C, to obtain the complete solution: -(1/16)cos(8x) + (1/4)cos(2x) + C. This process demonstrates the importance of trigonometric identities in simplifying integrals and the fundamental rules of integration in calculus. By following these steps, we can effectively solve similar trigonometric integrals. This systematic approach not only provides the correct answer but also enhances understanding of the underlying principles of calculus and trigonometric functions.
The process of evaluating the integral ∫ cos(5x) sin(3x) dx highlights the power and elegance of calculus techniques. By strategically applying the product-to-sum identity, we transformed a seemingly complex integral into a manageable one. This transformation is a prime example of how trigonometric identities can simplify mathematical problems. The step-by-step approach we followed, from applying the identity to integrating each term and adding the constant of integration, demonstrates the systematic nature of calculus. Each step was crucial in arriving at the final solution: -(1/16)cos(8x) + (1/4)cos(2x) + C. This solution represents a family of functions, differing only by a constant, whose derivative is the original integrand. The constant of integration, C, plays a vital role in capturing this family of functions. The successful evaluation of this integral showcases the importance of understanding and applying fundamental calculus principles. This includes not only the rules of integration but also the strategic use of trigonometric identities. The ability to recognize and apply these tools is essential for solving a wide range of problems in mathematics, physics, and engineering. The process we have demonstrated provides a clear and concise method for tackling similar trigonometric integrals. By mastering these techniques, students and professionals can confidently approach and solve complex calculus problems. The integral ∫ cos(5x) sin(3x) dx serves as a valuable example for learning and practicing these essential skills. The step-by-step methodology also helps in developing problem-solving strategies that are crucial for success in advanced mathematics courses.
