Integral Of 1 1 Sinx

Understanding and Solving the Integral of 1/(1+sin x)

The integral ∫1/(1+sin x) dx might seem deceptively simple at first glance, but it requires a clever trick to solve. This article will guide you through the process, breaking down the solution step-by-step and exploring the underlying mathematical principles. We'll cover various approaches, explain the rationale behind each step, and address frequently asked questions. This comprehensive guide will equip you with a solid understanding of this integral and similar trigonometric integrals.

Introduction: Why this Integral is Challenging and What We'll Cover

The seemingly straightforward integral, ∫1/(1+sin x) dx, presents a challenge because we cannot directly apply standard integration rules. Unlike simpler trigonometric integrals like ∫sin x dx or ∫cos x dx, this integral requires a strategic substitution to simplify the integrand. This article will explore several methods for solving this integral, focusing on a common and effective approach using a trigonometric identity and a substitution that transforms the integral into a manageable form. We’ll also delve into the mathematical reasoning behind each step, ensuring a comprehensive understanding of the process.

Method 1: Using the Weierstrass Substitution

The Weierstrass substitution is a powerful technique for solving integrals involving trigonometric functions. It involves substituting t = tan(x/2). This substitution transforms the trigonometric functions into rational functions of t, making them easier to integrate.

Steps:

1. The Substitution: Let t = tan(x/2). Then, sin x = 2t/(1+t²) and dx = 2/(1+t²) dt.

2. Substituting into the Integral: Replace sin x and dx in the original integral with their equivalents in terms of t:

   ∫1/(1+sin x) dx = ∫1/(1 + 2t/(1+t²)) * (2/(1+t²)) dt

3. Simplifying the Integrand: Combine the terms in the denominator:

   ∫2/(1+t² + 2t) dt = ∫2/(t²+2t+1) dt = ∫2/(t+1)² dt

4. Integrating: This is a simple power rule integration:

   2∫(t+1)^(-2) dt = 2 * [(t+1)^(-1) / -1] + C = -2/(t+1) + C

5. Back-Substituting: Replace t with tan(x/2):

   -2/(tan(x/2) + 1) + C

Therefore, the integral of 1/(1+sin x) is -2/(tan(x/2) + 1) + C, where C is the constant of integration.

Method 2: Multiplying by the Conjugate

Another effective method involves multiplying the numerator and denominator by the conjugate of the denominator. This approach cleverly eliminates the sine term from the denominator, making integration straightforward.

Steps:

1. Multiplying by the Conjugate: Multiply the numerator and denominator by (1-sin x):

   ∫1/(1+sin x) * (1-sin x)/(1-sin x) dx = ∫(1-sin x)/(1-sin²x) dx

2. Using a Trigonometric Identity: Recall the Pythagorean identity cos²x + sin²x = 1. Therefore, 1-sin²x = cos²x. Substitute this into the integral:

   ∫(1-sin x)/cos²x dx = ∫(1/cos²x - sin x/cos²x) dx

3. Separating the Integral: Split the integral into two simpler integrals:

   ∫(1/cos²x) dx - ∫(sin x/cos²x) dx

4. Integrating Each Part:

   • The first integral is a standard integral: ∫sec²x dx = tan x + C₁
   • The second integral can be solved using a u-substitution: Let u = cos x, then du = -sin x dx. This gives: ∫(sin x/cos²x) dx = -∫(1/u²) du = 1/u + C₂ = 1/cos x + C₂ = sec x + C₂

5. Combining the Results: Combining the results of the two integrations:

   tan x - sec x + C (where C = C₁ + C₂)

This result appears different from the one obtained using the Weierstrass substitution. However, both are correct and represent the same family of functions differing only by the constant of integration. The apparent discrepancy stems from the different forms these solutions take. Further manipulation of either solution can transform it into the other.

Mathematical Justification and Explanation of Steps

The success of both methods hinges on strategic manipulation of the integrand. The Weierstrass substitution cleverly transforms a trigonometric problem into an algebraic one, leveraging the properties of tangent half-angle formulas. Multiplying by the conjugate, on the other hand, cleverly uses algebraic manipulation to simplify the denominator using a fundamental trigonometric identity. The use of u-substitution in the second method highlights the power of substitution techniques in simplifying integration problems. In both cases, the resulting integrals are far more tractable than the original.

Addressing Potential Confusion and Common Errors

A common mistake is attempting to directly integrate 1/(1+sin x) using simple rules. This is incorrect because there isn't a direct antiderivative in the basic trigonometric functions. Another pitfall is making errors in algebraic simplification, particularly when dealing with trigonometric identities and fractional expressions. Careful attention to detail is crucial in ensuring accuracy at each step.

Frequently Asked Questions (FAQ)

• Q: Which method is better? A: Both methods are equally valid. The Weierstrass substitution is often preferred for its systematic approach to handling trigonometric integrals, while multiplying by the conjugate offers a more direct and intuitive approach for this specific problem. The best method depends on your familiarity and comfort level with different techniques.

• Q: What if the integral was ∫1/(1+cos x) dx? A: The methods are similar. You would either use the Weierstrass substitution or multiply by the conjugate (1-cos x). This will lead to a different, but equally valid, result.

• Q: How can I verify my answer? A: You can differentiate your final answer. If you obtain the original integrand, your integration is correct. Symbolic mathematics software can also help verify the result.

• Q: What is the significance of the constant of integration, C? A: The constant of integration represents an infinite family of functions, all of which have the same derivative. It accounts for the fact that the derivative of a constant is zero. The value of C is determined by the initial conditions or boundaries of the definite integral, if applicable.

Conclusion: Mastering Trigonometric Integration

Successfully solving ∫1/(1+sin x) dx requires a nuanced understanding of trigonometric identities and clever application of integration techniques. Whether you choose the Weierstrass substitution or the conjugate method, the key lies in transforming the integrand into a more manageable form. This problem serves as an excellent illustration of how seemingly complex integrals can be solved with strategic problem-solving techniques. Mastering such techniques is a crucial skill for anyone studying calculus or working with problems involving trigonometric functions. Remember, practice and patience are essential for developing proficiency in integration. By understanding the underlying principles and practicing different methods, you’ll become more confident and skilled in solving a wider range of challenging integrals.