Problem 56 - Thomas Calculus in SI Units

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int \frac{\csc \theta}{\csc \theta-\sin \theta} d \theta$$

Step-by-Step Solution

Answer

The most general antiderivative is \( \tan \theta + C \).

Step 1: Simplify the Integrand

The integrand is \( \frac{\csc \theta}{\csc \theta - \sin \theta} \). Begin by rewriting \( \csc \theta \) in terms of \( \sin \theta \), which gives us \( \frac{1}{\sin \theta} \). The integrand becomes: \[ \int \frac{1/\sin \theta}{1/\sin \theta - \sin \theta} \, d\theta = \int \frac{1}{1 - \sin^2 \theta} \, d\theta \]. Further simplify by noting that \( 1 - \sin^2 \theta = \cos^2 \theta \). Thus, the integrand simplifies to \( \frac{1}{\cos^2 \theta} = \sec^2 \theta \).

Step 2: Integrate the Simplified Expression

Now, integrate \( \sec^2 \theta \) with respect to \( \theta \). Recall the integral formula \( \int \sec^2 \theta \, d\theta = \tan \theta + C \), where \( C \) is the constant of integration. Thus, the antiderivative of \( \sec^2 \theta \) is \( \tan \theta + C \).

Step 3: Verify the Solution by Differentiation

Differentiate the solution \( \tan \theta + C \) to verify correctness. The derivative of \( \tan \theta \) is \( \sec^2 \theta \). Therefore, \( \frac{d}{d\theta}(\tan \theta + C) = \sec^2 \theta \), which matches the simplified integrand, confirming the antiderivative is correct.

Key Concepts

Antiderivative

An antiderivative or an indefinite integral is a fundamental concept in calculus involving finding a function whose derivative is the given function. In the exercise provided, we are asked to find the antiderivative of the function \( \frac{\csc \theta}{\csc \theta - \sin \theta} \). The process begins with expressing the given function in a simpler form to make integration feasible. By rewriting \( \csc \theta \) as \( \frac{1}{\sin \theta} \), and simplifying the expression step by step, the problem reduces to integrating \( \sec^2 \theta \).

When searching for an antiderivative, remember that there isn't a unique solution; it involves a constant of integration \( C \). This reflects the fact that derivatives of constants are zero, which means infinitely many functions with different constants can share the same derivative. To check if the found antiderivative is correct, differentiating it should yield the original function or integrand.

Trigonometric Integration

Trigonometric integration involves integrating functions that consist of trigonometric functions like sine, cosine, tangent, and their reciprocals. In this exercise, trigonometric integration is key for simplifying and solving the integral \( \int \frac{\csc \theta}{\csc \theta - \sin \theta} d\theta \). By recognizing the reciprocal identity \( \csc \theta = \frac{1}{\sin \theta} \), the expression can be rewritten and simplified using basic trigonometric identities.

• Firstly, an understanding of the Pythagorean identity \( 1 - \sin^2 \theta = \cos^2 \theta \) is utilized to convert the expression into \( \sec^2 \theta \).
• Finally, the integral of \( \sec^2 \theta \) is straightforward since \( \int \sec^2 \theta \, d\theta = \tan \theta + C \), where \( C \) is a constant of integration.

Recognizing when and how to apply these identities during integration is crucial for solving trigonometric integrals efficiently.

Calculus Problem Solving

Solving calculus problems often involves breaking down complex problems into simpler steps. The exercise provided showcases this in three distinct phases: simplifying the integrand, performing the integration, and verifying the solution through differentiation.

• Simplifying the Integrand: This is the initial step where algebraic manipulations and trigonometric identities are used to rewrite the integral. Here, trigonometric identities play a crucial role in transforming the problem into an easier form.
• Performing the Integration: With the simplified form, the integral becomes \( \int \sec^2 \theta \, d\theta \), which is well-known and can be computed directly.
• Verification: The final step involves differentiating the found antiderivative. If the derivative matches the original integrand, the solution is verified. This backward-check reinforces the correctness of the antiderivative.

By following these steps methodically, calculus problems become manageable and less daunting. Both analytical thinking and understanding of derivatives and integrals are essential for success.