Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of integrals, specifically tackling the integral J = ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx. This isn't just any integral; it's a fantastic example that combines trigonometric identities, clever algebraic manipulation, and a touch of calculus finesse. So, buckle up, grab your favorite beverage, and let's break it down together, guys!
Understanding the Challenge
Before we jump into the solution, let's appreciate the challenge this integral presents. The integrand, cos(2x) / (sin(x) + cos(x)), isn't something we can directly integrate using basic rules. It's a fraction with trigonometric functions in both the numerator and the denominator, hinting that we'll need to employ some strategic techniques to simplify it. The definite integral also adds another layer, as we will eventually have to evaluate the antiderivative at the limits of integration, 0 and π/4.
To effectively solve this, we'll be leveraging a powerful combination of trigonometric identities and algebraic manipulation. The key identity we'll be using is the double-angle formula for cosine: cos(2x) = cos²(x) - sin²(x). This identity will allow us to rewrite the numerator in a form that can be factored and potentially simplify the fraction. Furthermore, recognizing the potential for factorization will lead us to the difference of squares pattern, a crucial step in simplifying the expression. From there, we'll aim to cancel out terms and reduce the integral into a form that's easier to handle. The journey might seem complex, but with a clear strategy, we can conquer this integral, no problem!
Step 1: Unleashing the Power of Trigonometric Identities
The first step in our adventure is to simplify the integrand. As we discussed, the star of the show here is the double-angle identity for cosine:
cos(2x) = cos²(x) - sin²(x)
By substituting this into our integral, we get:
J = ∫₀^(π/4) (cos²(x) - sin²(x)) / (sin(x) + cos(x)) dx
Now, the numerator looks more promising! We have a difference of squares, which, as you might recall, can be factored. This is a significant breakthrough because it allows us to rewrite the integral in a more manageable form. The ability to recognize and apply trigonometric identities like this is a fundamental skill in calculus, and it's what transforms seemingly impossible integrals into solvable ones. So, always keep your trig identities handy, guys, they are your secret weapons!
Step 2: Factoring the Difference of Squares
Remember the difference of squares factorization? It states that:
a² - b² = (a + b)(a - b)
Applying this to our numerator, where a = cos(x) and b = sin(x), we get:
cos²(x) - sin²(x) = (cos(x) + sin(x))(cos(x) - sin(x))
Substituting this back into the integral, we now have:
J = ∫₀^(π/4) [(cos(x) + sin(x))(cos(x) - sin(x))] / (sin(x) + cos(x)) dx
Notice anything exciting? We have a common factor of (cos(x) + sin(x)) in both the numerator and the denominator! This is a beautiful moment in calculus, where our careful manipulation pays off with a significant simplification. The ability to spot these patterns and apply algebraic techniques like factoring is what makes problem-solving in calculus so rewarding. It's like detective work, where you uncover hidden structures and use them to crack the case, guys!
Step 3: The Sweetness of Cancellation
Now comes the satisfying part: cancellation! We can cancel the (cos(x) + sin(x)) term from both the numerator and the denominator, leaving us with:
J = ∫₀^(π/4) (cos(x) - sin(x)) dx
Wow! Look how much simpler the integral has become. We've transformed a complex fraction into a straightforward difference of trigonometric functions. This is a testament to the power of strategic simplification in calculus. By using trigonometric identities and algebraic manipulation, we've turned a seemingly daunting problem into a manageable one. The lesson here is clear: never be afraid to manipulate the expression, guys. It often leads to a much easier path.
Step 4: Integrating the Simplified Expression
Now that we have a simplified integral, the integration itself is relatively straightforward. We know the integrals of cos(x) and sin(x):
- ∫ cos(x) dx = sin(x) + C
- ∫ sin(x) dx = -cos(x) + C
Therefore, the integral of (cos(x) - sin(x)) is:
∫ (cos(x) - sin(x)) dx = sin(x) + cos(x) + C
So, we have the antiderivative. Now, we need to evaluate it at the limits of integration, π/4 and 0. Remember, the constant of integration, C, will cancel out when we evaluate the definite integral, so we can ignore it for now.
Step 5: Evaluating the Definite Integral
To evaluate the definite integral, we subtract the value of the antiderivative at the lower limit from its value at the upper limit:
J = [sin(π/4) + cos(π/4)] - [sin(0) + cos(0)]
We know the values of sine and cosine at these angles:
- sin(π/4) = √2 / 2
- cos(π/4) = √2 / 2
- sin(0) = 0
- cos(0) = 1
Substituting these values, we get:
J = [√2 / 2 + √2 / 2] - [0 + 1]
J = √2 - 1
Final Answer: The Grand Finale!
And there you have it, guys! The solution to the integral J = ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx is:
J = √2 - 1
We successfully navigated through the intricacies of this integral, using a combination of trigonometric identities, algebraic manipulation, and the fundamental principles of calculus. This problem highlights the importance of recognizing patterns, applying appropriate techniques, and breaking down complex problems into smaller, more manageable steps. Remember, practice makes perfect, so keep tackling those integrals, and you'll become a calculus pro in no time!
Let's embark on a journey to unravel the intricacies of the integral ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx. This integral serves as an exemplary illustration of how trigonometric identities, algebraic manipulations, and the fundamental principles of calculus converge to solve seemingly complex problems. Our approach will be methodical, breaking down each step to ensure clarity and comprehension. This detailed guide is designed to not only provide the solution but also to illuminate the underlying concepts and techniques involved, making it a valuable resource for students and enthusiasts alike. The focus will be on transforming the integrand into a form that is more amenable to integration, a common strategy in calculus that often involves a blend of creativity and mathematical rigor. By mastering such techniques, you'll be well-equipped to tackle a wider range of integral problems.
Unveiling the Challenge: A Multifaceted Approach
The integral ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx presents a unique challenge due to the nature of its integrand. Directly integrating the given expression is not feasible with elementary techniques, necessitating a strategic approach. The presence of trigonometric functions in both the numerator and the denominator suggests the application of trigonometric identities to simplify the expression. This simplification is often the key to unlocking the solution, as it transforms the integral into a more recognizable and integrable form. Moreover, the definite nature of the integral, with its specified limits of integration from 0 to π/4, adds another layer of complexity. It requires us not only to find the antiderivative but also to evaluate it at these limits, ultimately yielding a numerical value. The challenge integral isn't just about computation; it's about problem-solving, requiring us to identify the right tools and techniques to navigate the complexities of the expression. It's a puzzle that rewards careful analysis and strategic manipulation.
To effectively tackle this integral, we will need to employ a multi-faceted strategy. The cornerstone of our approach will be the use of the double-angle identity for cosine: cos(2x) = cos²(x) - sin²(x). This identity will allow us to rewrite the numerator in a form that can be factored, a crucial step in simplifying the fraction. By recognizing this opportunity for factorization, we can leverage the difference of squares pattern, a powerful tool in algebraic manipulation. This pattern will enable us to break down the numerator into factors that may potentially cancel with the denominator, leading to a simpler expression. The simplification process is not just about applying formulas; it's about developing a mathematical intuition, a sense for when and how to use different techniques to achieve the desired outcome. This intuition is built through practice and a deep understanding of the underlying principles. Once we've simplified the integrand, the integration itself will become more manageable, allowing us to find the antiderivative and evaluate it at the limits of integration.
Step 1: Harnessing Trigonometric Power: The Double-Angle Identity
Our initial move in this mathematical dance is to simplify the integrand using a fundamental trigonometric identity. As highlighted earlier, the double-angle identity for cosine serves as our primary tool:
cos(2x) = cos²(x) - sin²(x)
By substituting this identity into our integral, we transform it as follows:
∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx becomes ∫₀^(π/4) (cos²(x) - sin²(x)) / (sin(x) + cos(x)) dx
This substitution is more than just a cosmetic change; it's a strategic maneuver that unlocks a pathway to simplification. The numerator, now expressed as the difference of squares, reveals a hidden structure that we can exploit. This step underscores the importance of mastering trigonometric identities. They are not just isolated formulas; they are powerful tools that can transform complex expressions into simpler, more manageable forms. The ability to recognize and apply these identities is a hallmark of a skilled calculus practitioner. By rewriting the numerator in this way, we've set the stage for the next crucial step: factoring.
Step 2: The Art of Factoring: Unveiling Hidden Structures
Now, we leverage a fundamental algebraic principle: the difference of squares factorization. This principle states that:
a² - b² = (a + b)(a - b)
Applying this to our numerator, with a = cos(x) and b = sin(x), we obtain:
cos²(x) - sin²(x) = (cos(x) + sin(x))(cos(x) - sin(x))
Substituting this factorization back into the integral, we have:
∫₀^(π/4) [(cos(x) + sin(x))(cos(x) - sin(x))] / (sin(x) + cos(x)) dx
At this juncture, a significant simplification becomes apparent. We observe a common factor, (cos(x) + sin(x)), present in both the numerator and the denominator. This recognition is a critical moment in the problem-solving process. It demonstrates the power of algebraic manipulation in revealing hidden structures and paving the way for simplification. The ability to spot these common factors is not just about memorizing formulas; it's about developing a keen eye for mathematical patterns, a skill that is honed through practice and exposure to various problem-solving techniques. The next step, cancellation, will further streamline the integral, bringing us closer to the solution.
Step 3: The Elegance of Cancellation: Streamlining the Integral
The next step is both elegant and crucial: cancellation. We can gracefully cancel the common factor of (cos(x) + sin(x)) from both the numerator and the denominator, leading to a significantly simplified integral:
∫₀^(π/4) (cos(x) - sin(x)) dx
Observe the remarkable transformation! We've successfully converted a complex fractional expression into a simple difference of trigonometric functions. This dramatic simplification underscores the efficacy of our strategic manipulations. It highlights the power of combining trigonometric identities with algebraic techniques to reduce complexity. This step is not just about making the integral easier to solve; it's about gaining a deeper understanding of the mathematical structure and revealing the underlying simplicity. The simplified integral is now in a form that is readily integrable using basic calculus rules. This transformation is a testament to the beauty of mathematical simplification, where strategic steps can lead to dramatic results.
Step 4: Integrating with Confidence: Finding the Antiderivative
With the integral now in a simplified form, the integration process becomes straightforward. Recall the basic integrals of cosine and sine:
- ∫ cos(x) dx = sin(x) + C
- ∫ sin(x) dx = -cos(x) + C
Applying these rules, the integral of (cos(x) - sin(x)) is:
∫ (cos(x) - sin(x)) dx = sin(x) + cos(x) + C
Thus, we have successfully found the antiderivative. The constant of integration, C, is included for completeness, but it will ultimately cancel out when we evaluate the definite integral. This step emphasizes the importance of knowing the basic integration rules. They are the foundation upon which we build more complex integration techniques. The ability to quickly and accurately integrate common functions is essential for success in calculus. Now that we have the antiderivative, we proceed to the final step: evaluating it at the limits of integration.
Step 5: The Final Flourish: Evaluating the Definite Integral
To complete our journey, we must evaluate the definite integral. This involves subtracting the value of the antiderivative at the lower limit of integration from its value at the upper limit:
[sin(π/4) + cos(π/4)] - [sin(0) + cos(0)]
We know the values of sine and cosine at these key angles:
- sin(π/4) = √2 / 2
- cos(π/4) = √2 / 2
- sin(0) = 0
- cos(0) = 1
Substituting these values, we get:
[√2 / 2 + √2 / 2] - [0 + 1] = √2 - 1
This evaluation brings us to the final numerical answer. It's a culmination of all the previous steps, demonstrating the power of calculus in providing precise solutions to mathematical problems. The ability to accurately evaluate trigonometric functions at key angles is a crucial skill, and it's often necessary for solving definite integrals. This final step is not just about plugging in numbers; it's about bringing together all the pieces of the puzzle and arriving at the solution.
The Grand Finale: The Solution Unveiled
Therefore, the solution to the integral ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx is:
√2 - 1
We have successfully navigated the complexities of this integral, employing a strategic combination of trigonometric identities, algebraic manipulation, and the fundamental principles of calculus. This problem serves as a valuable lesson in the importance of careful analysis, strategic thinking, and the power of simplification. Remember, practice and persistence are key to mastering calculus, so keep exploring, keep questioning, and keep solving!
This article will guide you through the solution of the integral ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx, a fascinating problem that showcases the elegance and power of calculus. We'll break down the solution step-by-step, highlighting the key techniques and concepts involved. This integral is a great example of how trigonometric identities and algebraic manipulation can be used to simplify seemingly complex expressions, making them amenable to integration. By understanding the process outlined here, you'll gain valuable insights into the art of problem-solving in calculus, developing skills that can be applied to a wide range of mathematical challenges. We will focus not just on arriving at the answer, but on understanding the why behind each step, fostering a deeper appreciation for the mathematical principles at play. So, let's dive in and unlock the secrets of this integral together!
The Integral Challenge: A Road Map to Success
The integral ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx presents a multi-layered challenge. The integrand itself, a fraction involving trigonometric functions, isn't directly integrable using standard techniques. This necessitates a strategic approach, one that involves transforming the integrand into a more manageable form. The presence of cos(2x) in the numerator hints at the potential application of the double-angle formula, a crucial trigonometric identity that can help us rewrite the expression. Moreover, the denominator, sin(x) + cos(x), suggests that we might be able to factor the numerator in a way that allows for cancellation, a common and powerful simplification technique. The definite integral nature of the problem, with its specific limits of integration, adds another layer of complexity. We'll need to find the antiderivative of the simplified integrand and then evaluate it at these limits, ultimately arriving at a numerical result. Successfully tackling this integral requires a blend of trigonometric knowledge, algebraic skill, and a solid understanding of the fundamental theorem of calculus. It's a problem that rewards careful planning and a step-by-step approach.
Our strategy for solving this integral will revolve around a few key ideas. First and foremost, we'll leverage the double-angle identity for cosine, cos(2x) = cos²(x) - sin²(x), to rewrite the numerator. This transformation is crucial because it opens the door for factoring, a powerful algebraic technique that can dramatically simplify expressions. By recognizing the difference of squares pattern, we can factor the numerator into (cos(x) + sin(x))(cos(x) - sin(x)). This factorization is the key to unlocking the solution, as it reveals a common factor with the denominator. The ability to spot these patterns and apply appropriate techniques is a hallmark of mathematical proficiency. It's about recognizing the underlying structure of the problem and using your tools to unravel its complexities. Once we've factored the numerator, we can cancel the common factor with the denominator, simplifying the integrand significantly. From there, the integration process becomes much more manageable, allowing us to find the antiderivative and evaluate it at the limits of integration. This strategic approach, combining trigonometric identities and algebraic manipulation, is a powerful paradigm for solving a wide range of calculus problems.
Step 1: Employing Trigonometry: The Double-Angle Identity in Action
The first step towards solving this integral involves harnessing the power of trigonometry. Specifically, we'll utilize the double-angle identity for cosine:
cos(2x) = cos²(x) - sin²(x)
Substituting this identity into the original integral, we obtain:
∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx becomes ∫₀^(π/4) (cos²(x) - sin²(x)) / (sin(x) + cos(x)) dx
This substitution is a strategic move, transforming the numerator into a form that we can further manipulate. The expression cos²(x) - sin²(x) is a classic example of a difference of squares, a pattern that can be factored. This is a crucial observation, as factoring will allow us to simplify the integrand and potentially cancel terms. This step underscores the importance of having a strong foundation in trigonometric identities. They are not just formulas to memorize; they are powerful tools that can unlock hidden structures within mathematical expressions. By recognizing the applicability of the double-angle identity, we've taken a significant step towards solving the integral. The next step, factoring, will further simplify the expression, bringing us closer to the solution.
Step 2: Unleashing Algebra: Factoring the Difference of Squares
Building upon our previous step, we now employ a fundamental algebraic technique: factoring the difference of squares. Recall the general formula:
a² - b² = (a + b)(a - b)
Applying this to our numerator, with a = cos(x) and b = sin(x), we get:
cos²(x) - sin²(x) = (cos(x) + sin(x))(cos(x) - sin(x))
Substituting this factorization back into the integral, we have:
∫₀^(π/4) [(cos(x) + sin(x))(cos(x) - sin(x))] / (sin(x) + cos(x)) dx
Now, the power of factoring becomes evident. We see a common factor, (cos(x) + sin(x)), appearing in both the numerator and the denominator. This is a key observation, as it allows us to simplify the expression through cancellation. This step highlights the interplay between algebra and calculus. Algebraic manipulation, such as factoring, is often essential for simplifying expressions before applying calculus techniques. The ability to recognize and apply these algebraic tools is a crucial skill for any calculus student. By factoring the numerator, we've paved the way for the next step, cancellation, which will dramatically simplify the integral.
Step 3: The Magic of Simplification: Cancellation and Integral Transformation
The next step is a moment of mathematical elegance: cancellation. We can cancel the common factor of (cos(x) + sin(x)) from both the numerator and the denominator, resulting in the simplified integral:
∫₀^(π/4) (cos(x) - sin(x)) dx
The transformation is remarkable! A complex fractional integrand has been reduced to a simple difference of trigonometric functions. This simplification underscores the effectiveness of our strategy, highlighting the power of combining trigonometric identities and algebraic manipulation. The integral is now in a form that is easily integrable using basic calculus rules. This step is a testament to the beauty of mathematical problem-solving. By strategically applying the right techniques, we can transform seemingly daunting problems into manageable ones. The simplified integral now sets the stage for the next step: finding the antiderivative.
Step 4: Calculus in Action: Finding the Antiderivative
With the integral simplified, we can now proceed with the integration itself. Recall the fundamental integrals of cosine and sine:
- ∫ cos(x) dx = sin(x) + C
- ∫ sin(x) dx = -cos(x) + C
Applying these rules, the integral of (cos(x) - sin(x)) is:
∫ (cos(x) - sin(x)) dx = sin(x) + cos(x) + C
Thus, we have found the antiderivative. The constant of integration, C, is included for completeness, but it will cancel out when we evaluate the definite integral. This step emphasizes the importance of knowing the basic integration rules. They are the building blocks for solving more complex integrals. The ability to quickly and accurately find the antiderivatives of common functions is essential for success in calculus. Now that we have the antiderivative, we move on to the final step: evaluating it at the limits of integration.
Step 5: The Grand Finale: Evaluating the Definite Integral and Unveiling the Solution
To complete our journey, we must evaluate the definite integral. This involves substituting the limits of integration into the antiderivative and subtracting the results:
[sin(π/4) + cos(π/4)] - [sin(0) + cos(0)]
We recall the values of sine and cosine at these key angles:
- sin(π/4) = √2 / 2
- cos(π/4) = √2 / 2
- sin(0) = 0
- cos(0) = 1
Substituting these values, we get:
[√2 / 2 + √2 / 2] - [0 + 1] = √2 - 1
This final calculation reveals the solution to the integral. It's the culmination of all our previous efforts, demonstrating the power of calculus in providing precise answers to mathematical questions. This step is not just about arithmetic; it's about bringing together all the pieces of the puzzle and arriving at the final result. The ability to accurately evaluate trigonometric functions and perform basic arithmetic is essential for completing the solution.
The Ultimate Answer: The Integral's Value Revealed
Therefore, the value of the integral ∫₀^(π/4) cos(2x) / (sin(x) + cos(x)) dx is:
√2 - 1
We have successfully navigated the intricacies of this integral, employing a strategic combination of trigonometric identities, algebraic manipulation, and the fundamental principles of calculus. This problem serves as a valuable lesson in the importance of careful analysis, strategic thinking, and the power of simplification. By understanding the steps outlined here, you'll be well-equipped to tackle similar challenges in calculus. Remember, practice is key to mastering these techniques, so keep exploring, keep questioning, and keep solving!