Evaluating The Integral Of 2/(1 + Sin X) From Π/6 To Π/3

This article delves into the step-by-step evaluation of the definite integral ∫(π/6)^(π/3) [2 / (1 + sin x)] dx. This type of integral, involving trigonometric functions, often appears in calculus and requires specific techniques to solve. We will explore the use of trigonometric identities and substitutions to simplify the integrand and find the definite integral over the given limits. This comprehensive guide will benefit students, educators, and anyone interested in mastering integral calculus techniques.

1. Introduction to the Integral

The core of this exploration is the definite integral ∫(π/6)^(π/3) [2 / (1 + sin x)] dx. This mathematical expression represents the area under the curve of the function f(x) = 2 / (1 + sin x) between the limits x = π/6 and x = π/3. To find this area, we need to evaluate the integral. Integrals involving trigonometric functions like sine and cosine often require clever manipulations to make them solvable. The presence of 1 + sin x in the denominator suggests that we might need to use trigonometric identities or a suitable substitution to simplify the expression. Before diving into the calculations, it's crucial to understand the behavior of the function we are integrating. The function 2 / (1 + sin x) is continuous and positive in the interval [π/6, π/3], which makes the integral well-defined. The limits of integration, π/6 and π/3, are standard angles where the sine function has known values, which will be helpful during the evaluation process. This integral serves as a good example for demonstrating how trigonometric identities and substitutions can be applied to solve seemingly complex integrals. By working through this example, we can gain valuable insights into handling similar integrals in calculus. The approach we take will involve rationalizing the denominator, which is a common technique when dealing with expressions involving 1 + sin x or 1 + cos x in the denominator. The goal is to transform the integral into a form that is easier to integrate using standard techniques. Understanding the nuances of this process is vital for anyone studying calculus and its applications in various fields of science and engineering. As we proceed, we will carefully explain each step, ensuring clarity and understanding for the reader. Let's now move on to the first step in solving this integral: rationalizing the denominator.

2. Rationalizing the Denominator

The initial hurdle in evaluating the integral ∫(π/6)^(π/3) [2 / (1 + sin x)] dx is the presence of 1 + sin x in the denominator. A common strategy to tackle such expressions is to rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator. In this case, the conjugate of 1 + sin x is 1 - sin x. By multiplying both the numerator and the denominator by 1 - sin x, we aim to eliminate the sine function from the denominator, making the integral easier to handle. This process utilizes the trigonometric identity sin^2(x) + cos^2(x) = 1, which will help simplify the denominator into a squared term. When we multiply the denominator (1 + sin x) by its conjugate (1 - sin x), we get 1 - sin^2(x). Using the Pythagorean identity, 1 - sin^2(x) is equivalent to cos^2(x). This transformation is a key step in simplifying the integral. Now, let's perform the multiplication: [2 / (1 + sin x)] * [(1 - sin x) / (1 - sin x)] = [2(1 - sin x)] / [1 - sin^2(x)] = [2(1 - sin x)] / cos^2(x). This manipulation has successfully transformed the denominator into a single trigonometric function, cos^2(x). The next step is to separate the fraction into two simpler fractions, each involving only one trigonometric function. This will allow us to use standard integral formulas to evaluate the integral. The rationalization of the denominator is a powerful technique that can be applied to a variety of integrals involving trigonometric functions. It helps in transforming complex expressions into simpler forms, making the integration process more manageable. Understanding this technique is essential for solving a wide range of calculus problems. Let's now move on to the next step, where we separate the fraction and simplify it further to make it amenable to integration. This will involve breaking down the expression into terms that we can directly integrate using standard integral formulas.

3. Separating and Simplifying the Integrand

After rationalizing the denominator, our integral has been transformed into ∫(π/6)^(π/3) [2(1 - sin x) / cos^2(x)] dx. Now, the next logical step is to separate the integrand into two simpler fractions. This allows us to deal with each term individually, making the integration process more manageable. We can rewrite the integrand as: [2(1 - sin x) / cos^2(x)] = 2[1 / cos^2(x) - sin x / cos^2(x)]. Now, we have two terms: 1 / cos^2(x) and sin x / cos^2(x). We recognize that 1 / cos^2(x) is the same as sec^2(x). For the second term, sin x / cos^2(x), we can rewrite it as (sin x / cos x) * (1 / cos x), which is equivalent to tan x * sec x. So, our integrand becomes: 2[sec^2(x) - tan x * sec x]. Now, the integral can be written as: ∫(π/6)^(π/3) 2[sec^2(x) - tan x * sec x] dx. We can further simplify this by distributing the constant 2: 2 ∫(π/6)^(π/3) [sec^2(x) - tan x * sec x] dx. This form is much easier to integrate because we know the antiderivatives of sec^2(x) and tan x * sec x. The antiderivative of sec^2(x) is tan x, and the antiderivative of tan x * sec x is sec x. Separating and simplifying the integrand is a common technique in calculus. It allows us to break down complex expressions into simpler components that are easier to integrate. Recognizing trigonometric identities and knowing the derivatives and antiderivatives of trigonometric functions are crucial skills for this step. By applying these techniques, we have successfully transformed the integral into a form that we can now directly integrate. In the next section, we will find the antiderivative and evaluate it at the limits of integration to find the definite integral.

4. Finding the Antiderivative

Having simplified the integral to 2 ∫(π/6)^(π/3) [sec^2(x) - tan x * sec x] dx, we now focus on finding the antiderivative of the integrand. As mentioned earlier, the antiderivative of sec^2(x) is tan x, and the antiderivative of tan x * sec x is sec x. Therefore, the antiderivative of sec^2(x) - tan x * sec x is tan x - sec x. Since we have a constant factor of 2 outside the integral, we multiply the antiderivative by 2. Thus, the antiderivative of the entire integrand is 2(tan x - sec x). This step involves recalling the fundamental rules of integration and the derivatives of trigonometric functions. The ability to quickly recognize these antiderivatives is essential for efficient problem-solving in calculus. It's also important to remember that when finding the antiderivative, we are essentially reversing the process of differentiation. So, if we were to differentiate 2(tan x - sec x), we should get back our original integrand, 2[sec^2(x) - tan x * sec x]. This is a good way to check our work and ensure that we have found the correct antiderivative. The next step is to evaluate this antiderivative at the limits of integration, π/3 and π/6. This will give us the value of the definite integral. Evaluating the antiderivative at the limits involves substituting the upper and lower limits into the antiderivative and subtracting the results. This process gives us the net change in the antiderivative over the interval of integration, which is the value of the definite integral. Finding the antiderivative is a crucial step in evaluating definite integrals. It requires a solid understanding of differentiation rules and the ability to recognize patterns. With the antiderivative in hand, we are now ready to evaluate the integral at the given limits and find the final answer.

5. Evaluating at the Limits of Integration

Now that we have found the antiderivative, 2(tan x - sec x), the final step is to evaluate it at the limits of integration, π/3 and π/6. This involves substituting these values into the antiderivative and subtracting the result at the lower limit from the result at the upper limit. So, we need to calculate: 2[tan(π/3) - sec(π/3)] - 2[tan(π/6) - sec(π/6)]. First, let's find the values of the trigonometric functions at these angles. We know that: tan(π/3) = √3 sec(π/3) = 2 tan(π/6) = 1/√3 = √3/3 sec(π/6) = 2/√3 = 2√3/3 Now, substitute these values into the expression: 2[(√3) - 2] - 2[(√3/3) - (2√3/3)] = 2(√3 - 2) - 2(-√3/3) = 2√3 - 4 + (2√3/3) To simplify further, we can combine the terms with √3: = 2√3 + (2√3/3) - 4 = (6√3 + 2√3) / 3 - 4 = (8√3/3) - 4 So, the value of the definite integral is (8√3/3) - 4. This is the final answer, representing the area under the curve of the function 2 / (1 + sin x) between the limits x = π/6 and x = π/3. Evaluating at the limits of integration is a crucial step in finding the definite integral. It involves careful substitution and simplification to arrive at the final numerical value. Understanding the values of trigonometric functions at standard angles is essential for this step. This process demonstrates the power of calculus in finding areas and solving problems involving continuous functions. By following these steps, we have successfully evaluated the definite integral and gained a deeper understanding of the techniques involved in integral calculus. Let's recap the entire process and highlight the key steps in the conclusion.

6. Conclusion

In conclusion, we have successfully evaluated the definite integral ∫(π/6)^(π/3) [2 / (1 + sin x)] dx. The process involved several key steps: rationalizing the denominator, separating and simplifying the integrand, finding the antiderivative, and evaluating at the limits of integration. First, we rationalized the denominator by multiplying both the numerator and the denominator by the conjugate of 1 + sin x, which is 1 - sin x. This transformed the integral into a form with cos^2(x) in the denominator. Next, we separated the integrand into two simpler fractions, sec^2(x) and tan x * sec x, which allowed us to use standard integral formulas. We then found the antiderivative of sec^2(x) - tan x * sec x, which is tan x - sec x. Multiplying by the constant factor 2, we got the antiderivative as 2(tan x - sec x). Finally, we evaluated this antiderivative at the limits of integration, π/3 and π/6, and simplified the result to obtain the value of the definite integral as (8√3/3) - 4. This example showcases the importance of trigonometric identities and substitutions in solving integrals involving trigonometric functions. The ability to recognize patterns and apply appropriate techniques is crucial for mastering integral calculus. The process also highlights the fundamental theorem of calculus, which connects differentiation and integration. By finding the antiderivative and evaluating it at the limits of integration, we can determine the area under the curve of a function over a given interval. This skill is essential in various fields of science, engineering, and mathematics. Through this step-by-step guide, we have demonstrated how to approach and solve a challenging integral, providing a valuable resource for students and anyone interested in calculus. The techniques used here can be applied to a wide range of similar problems, making this a valuable learning experience. By understanding and practicing these methods, one can gain confidence in tackling complex integrals and further explore the fascinating world of calculus. The journey through this integral has reinforced the power and elegance of calculus in solving real-world problems and advancing our understanding of mathematical concepts.