Problem 41 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int \frac{\csc \theta \cot \theta}{2} d \theta $$

Step-by-Step Solution

Answer

The most general antiderivative is \( \frac{-1}{2} \csc \theta + C \).

Step 1: Recognize the Structure of the Integral

Notice that the integral is \( \int \frac{\csc \theta \cot \theta}{2} d \theta \). The integrand is in a form that resembles the derivative of a standard trigonometric function.

Step 2: Simplify and Identify the Antiderivative

Recall that the derivative of \( \csc \theta \) is \( -\csc \theta \cot \theta \). Thus, the antiderivative of \( \csc \theta \cot \theta \) is \( -\csc \theta \). Since we have \( \frac{1}{2} \) as a factor, we can factor it out: \[ \int \frac{\csc \theta \cot \theta}{2} d \theta = \frac{1}{2} \int \csc \theta \cot \theta \, d \theta = \frac{-1}{2} \csc \theta + C \] where \( C \) is the constant of integration.

Step 3: Differentiate to Check the Result

Now we differentiate the function \( \frac{-1}{2} \csc \theta + C \) to verify our antiderivative. The derivative is: \[ \frac{d}{d\theta} \left( \frac{-1}{2} \csc \theta + C \right) = \frac{-1}{2} (-\csc \theta \cot \theta) = \frac{1}{2} \csc \theta \cot \theta \] which matches the original integrand, confirming that our antiderivative is correct.

Key Concepts

Trigonometric Integrals

Trigonometric integrals are integrals that involve trigonometric functions like sine, cosine, and tangent, among others. These integrals often require knowledge of basic trigonometric identities, as well as derivatives of these functions. For instance, knowing that the derivative of \(\csc \theta\) is \(-\csc \theta \cot \theta\) helps identify that the antiderivative of \(\csc \theta \cot \theta\) is \(-\csc \theta\). This knowledge is essential when solving integrals involving trigonometric expressions.
Trigonometric integrals are also important in calculus as they appear in various problems, including those in physics and engineering. Understanding these integrals allows students to solve complex problems involving wave functions, oscillations, and other periodic phenomena. Therefore, solid knowledge of trigonometric functions and their properties is vital when working with these types of integrals.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us how the function's value changes as its input changes. In the context of verifying an antiderivative, differentiation is a crucial step. After finding the antiderivative, as was done in the solution with \(\frac{-1}{2} \csc \theta + C \), we use differentiation to confirm that when we differentiate this function, we get back the original integrand. This step ensures that the function we found is indeed the correct antiderivative.
The power of differentiation lies in its ability to analyze functions' behaviors graphically and numerically. It helps determine slope, rates of change, and can even help find maxima and minima of functions. In calculus, differentiating trigonometric functions is a frequent task, requiring knowledge of various differentiation rules and identities. By regularly practicing differentiation, students can become adept at verifying the solutions to integrals and understanding the dynamic relationships between functions.

Constant of Integration

Anytime you find an indefinite integral (antiderivative), you will see a \( C \), known as the constant of integration. This constant is critical because it represents an infinite set of possible solutions. When we integrate, the original function might have a constant term that disappears when differentiated to get the integrand, making us miss it when only reversing the process through integration.
By adding \( C \), we account for all possible vertical translations of the antiderivative graph. Consider the antiderivative \( \frac{-1}{2} \csc \theta + C \); this expression represents not one but multiple functions, each corresponding to a different constant \( C \). This concept emphasizes the importance of accurately including and understanding constants when working on indefinite integrals, as it provides a full family of solutions.