Problem 54 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. 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 \( \frac{\csc \theta}{\csc \theta - \sin \theta} \). Start by rewriting \( \csc \theta \) in terms of \( \sin \theta \):\[ \csc \theta = \frac{1}{\sin \theta} . \] The integrand becomes \[ \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} - \sin \theta} = \frac{1}{1 - \sin^2 \theta} . \] Recall 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 Function

Now, integrate the simplified function. We have \( \int \sec^2 \theta \,d\theta \). The antiderivative of \( \sec^2 \theta \) is a well-known result: \[ \int \sec^2 \theta \, d\theta = \tan \theta + C , \] where \( C \) is the constant of integration.

Step 3 - Verify by Differentiation

Verify the result by differentiating \( \tan \theta + C \). The derivative of \( \tan \theta \) is \( \sec^2 \theta \), and the derivative of a constant \( C \) is zero. Thus, differentiating \( \tan \theta + C \) gives \( \sec^2 \theta \), which matches our original simplified integrand, confirming our solution is correct.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve functions like sine, cosine, tangent, and their reciprocals. In the exercise, we had the integral of a trigonometric expression, specifically the cosecant (csc) function. Understanding trigonometric identities is fundamental.
For instance, knowing

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( 1 - \sin^2 \theta = \cos^2 \theta \)

is important. This knowledge allows us to simplify the integrand properly.
In trigonometric integrals, simplification often involves rewriting functions using these identities. This process makes it possible to use standard antiderivatives that are easier to integrate. Hence, simplifying the function to \( \sec^2 \theta \) using basic trigonometric identities, transforms a potentially complex problem into a straightforward solution. It's this simplification that ultimately makes integrating such expressions possible.

Indefinite Integrals

Indefinite integrals represent the collection of antiderivatives for a function. They offer a family of functions who's derivative would return to the original function. In our task, we determined the indefinite integral\[ \int \sec^2 \theta \, d\theta = \tan \theta + C\]where \(C\) is the constant of integration.
The constant of integration is essential, as indefinite integrals have more than one solution. Adding \(C\) acknowledges these different scenarios. This function, \(\tan \theta + C\), accounts for all possible vertical shifts of \(\tan \theta \), still having the correct derivative. This is what gives integrals their defining characteristic of a general solution. The indefinite integral is central as it helps connect different areas of calculus, facilitating the transition from derivatives back to original functions.

Calculus

Calculus is the study of how things change. It has two main branches: differentiation and integration. Differentiation focuses on rates of change, while integration centers on accumulation of quantities.
In this exercise, integration found a general solution to a trigonometric integral. We used simplification to turn the integration into a form that matches a known antiderivative. We can be certain of our solution by checking it through differentiation.
By differentiating the result (\(\tan \theta + C\)), you can confirm it leads back to the original function (\(\sec^2 \theta\)). This technique bridges the different aspects of calculus, ensuring precision and reliability in problem-solving. Understanding these connections enhances your skills in solving a wide range of mathematical problems, making calculus a versatile and powerful tool.