Integral Of 1 1 Sinx

Understanding and Solving the Integral of 1/(1 + sinx)

The integral of 1/(1 + sinx) dx is a classic example of a trigonometric integral that requires a bit of clever manipulation to solve. It's not immediately obvious how to approach this, but with the right techniques, we can break it down into a manageable form. This article will guide you through the process, explaining not just the steps involved but also the underlying mathematical principles. We'll cover various methods, discuss their advantages and disadvantages, and even address some frequently asked questions. This comprehensive guide aims to equip you with a thorough understanding of this integral and similar trigonometric integrations.

Introduction: Why This Integral Matters

The integral ∫ 1/(1 + sinx) dx appears frequently in calculus courses and various applications in physics and engineering. It showcases several important integration techniques, including trigonometric substitutions and the use of clever algebraic manipulations to simplify complex expressions. Mastering this integral builds a strong foundation for tackling more intricate trigonometric integrals. Understanding the various solution methods enhances your problem-solving skills and deepens your comprehension of integral calculus.

Method 1: Using the Weierstrass Substitution (t-substitution)

This method is often preferred for its systematic approach and relative ease of understanding. It involves a substitution that transforms the trigonometric function into a rational function, which is generally easier to integrate.

Steps:

1. The Substitution: We use the Weierstrass substitution: t = tan(x/2). This implies that:

   • sinx = 2t/(1 + t²)
   • cosx = (1 - t²)/(1 + t²)
   • dx = 2/(1 + t²) dt

2. Substituting into the Integral: Substitute the expressions for sinx and dx into the original integral:

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

3. Simplifying the Integrand: Simplify the expression:

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

4. Integrating: Now, we have a simple integral to solve:

∫ 2/( (t + 1)² ) dt = -2/(t + 1) + C

5. Back-substituting: Finally, substitute back t = tan(x/2):

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

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

Method 2: Multiplying by the Conjugate

This method involves a clever algebraic trick to simplify the integrand. It's less systematic than the Weierstrass substitution but can be quicker for those comfortable manipulating trigonometric identities.

Steps:

1. Multiplying by the Conjugate: Multiply the numerator and denominator by the conjugate of the denominator (1 - sinx):

∫ 1/(1 + sinx) dx = ∫ (1 - sinx)/((1 + sinx)(1 - sinx)) dx

2. Simplifying the Denominator: Using the difference of squares, simplify the denominator:

(1 + sinx)(1 - sinx) = 1 - sin²x = cos²x

3. Simplifying the Integrand: The integral now becomes:

∫ (1 - sinx)/cos²x dx = ∫ (1/cos²x - sinx/cos²x) dx

4. Separating the Integral: Separate the integral into two simpler integrals:

∫ (1/cos²x) dx - ∫ (sinx/cos²x) dx

5. Integrating: These integrals are standard:

• ∫ (1/cos²x) dx = tanx + C₁
• ∫ (sinx/cos²x) dx = 1/cosx + C₂ (This can be solved using u-substitution, with u = cosx)

6. Combining the Results: Combine the results to obtain the final answer:

tanx - 1/cosx + C (where C = C₁ + C₂)

Therefore, the integral of 1/(1 + sinx) dx is tanx - secx + C, where C is the constant of integration.

Comparing Methods 1 and 2

Both methods yield the same result, although they appear different at first glance. The Weierstrass substitution provides a more general and systematic approach suitable for a wider range of trigonometric integrals. The conjugate method is sometimes faster for this specific integral but might not be as adaptable to other problems. The apparent difference in the answers stems from the trigonometric identities connecting tan(x/2) and the trigonometric functions of x. A careful application of those identities will demonstrate the equivalence of the two results.

The Scientific Explanation and Underlying Principles

The success of both methods relies heavily on our understanding of trigonometric identities and substitution techniques in integration. The Weierstrass substitution cleverly transforms a trigonometric integral into a rational integral, a simpler form often easier to manage. Multiplying by the conjugate utilizes the difference of squares identity to simplify the integrand, making it directly integrable using known standard integrals. Both methods demonstrate the power of algebraic manipulation and strategic substitution in solving complex integrals. The core principle is to transform the integral into a form we can readily integrate using known integration rules and formulas.

Frequently Asked Questions (FAQ)

• Q: Why are there two different-looking answers? A: While the two methods produce seemingly different answers (-2/(tan(x/2) + 1) + C and tanx - secx + C), they are equivalent. This is due to trigonometric identities relating tan(x/2) to tanx and secx. A careful application of these identities will prove their equivalence.

• Q: Which method is better? A: The Weierstrass substitution is generally more versatile and applicable to a broader range of trigonometric integrals. The conjugate method is quicker for this specific problem but might not generalize as well. Choose the method that best suits your understanding and the specific problem at hand.

• Q: What if the integral was 1/(1 - sinx)? A: The process is very similar. You can either use the Weierstrass substitution or multiply by the conjugate (1 + sinx). The resulting integral will be slightly different, but the techniques remain the same.

• Q: What is the constant of integration, C, and why is it important? A: The constant of integration, C, represents an arbitrary constant added to the result of any indefinite integral. It's crucial because the derivative of a constant is zero. Therefore, infinitely many functions can have the same derivative. The constant C accounts for this ambiguity.

• Q: How can I check my answer? A: You can check your answer by differentiating the result. The derivative of your answer should be equal to the original integrand, 1/(1 + sinx).

Conclusion: Mastering Trigonometric Integration

This article provided a detailed exploration of the integral of 1/(1 + sinx) dx, employing two different yet equally valid methods. We've gone beyond simply presenting the solution, emphasizing the underlying mathematical principles and the reasoning behind each step. Mastering integrals like this one requires not only procedural knowledge but also a deep understanding of trigonometric identities, substitution techniques, and algebraic manipulation. By understanding both methods and their nuances, you build a stronger foundation for tackling more complex problems in integral calculus and related fields. Remember to always practice and explore different approaches to solidify your understanding and build your problem-solving skills. The beauty of mathematics lies in its elegance and the interconnectedness of its concepts; by mastering these techniques, you unlock a deeper appreciation of this elegant field.