Problem 49 - University Calculus: Early Transcendentals

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. $$\int \frac{\csc \theta \cot \theta}{2} d \theta$$

Step-by-Step Solution

Answer

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

Step 1: Identify the integral form

The given integral is \( \int \frac{\csc \theta \cot \theta}{2} \, d\theta \). We can rewrite it as \( \int \frac{1}{2} \csc \theta \cot \theta \, d\theta \), which helps in focusing on the integration of the trigonometric expression \( \csc \theta \cot \theta \).

Step 2: Recognize the derivative

Recall that the derivative of \( \csc \theta \) is \( -\csc \theta \cot \theta \). Thus, noticing the connection, we can consider that the integral of \( \csc \theta \cot \theta \) is related to \( \csc \theta \).

Step 3: Integrate the trigonometric function

Using the recognition from the previous step, it follows that the integral of \( \csc \theta \cot \theta \) is \( -\csc \theta \). Therefore, the integral \( \int \frac{1}{2} \csc \theta \cot \theta \, d\theta \) is equal to \( -\frac{1}{2} \csc \theta + C \), where \( C \) is the constant of integration.

Step 4: Verify by differentiation

Differentiate \( -\frac{1}{2} \csc \theta + C \) to check if it returns to the original integrand. The derivative is \( \frac{1}{2} \csc \theta \cot \theta \), which matches the given integrand \( \frac{\csc \theta \cot \theta}{2} \). Thus, our solution is verified.

Key Concepts

Trigonometric Integration

Trigonometric integration is a technique used to solve integrals that involve trigonometric functions. It often requires recognizing patterns and identities to simplify or manipulate expressions into more integrable forms. In this exercise, the integral \( \int \frac{\csc \theta \cot \theta}{2} d\theta \) involves the trigonometric functions cosecant \( \csc \theta \) and cotangent \( \cot \theta \). Approach this integral by rewriting it as \( \int \frac{1}{2} \csc \theta \cot \theta \, d\theta \). This method helps focus on the trigonometric function itself, allowing us to use known derivatives or integration techniques. Given that the derivative of \( \csc \theta \) is \( -\csc \theta \cot \theta \), the integral can be directly linked, showing that our initial function corresponds well with recognizable patterns. By improving your familiarity with common derivatives and antiderivatives of trigonometric functions, you can efficiently solve these integrals.

Antiderivatives

An antiderivative is a function whose derivative gives back the original function. In the context of indefinite integrals, finding an antiderivative means determining the most general form of the original function. Indefinite integrals include a constant of integration, \( C \), representing the infinite set of all possible antiderivatives. To solve \( \int \frac{1}{2} \csc \theta \cot \theta \, d\theta \), focus on recognizing that the structure resembles the known derivative of \( \csc \theta \), given by \( -\csc \theta \cot \theta \). Therefore, the antiderivative can be quickly identified as \( -\frac{1}{2} \csc \theta + C \). This represents the most general form because the constant \( C \) accounts for any vertical shift possible in the family of solutions. Understanding this fundamental concept allows us to identify how particular functions are reversely constructed through differentiation.

Differentiation Verification

Differentiation verification is a crucial step in confirming that a found antiderivative is correct. It involves differentiating the obtained antiderivative to ensure it simplifies back to the original integrand. This process acts as a check against miscalculations or incorrect assumptions made during integration.For the integral \( \int \frac{1}{2} \csc \theta \cot \theta \, d\theta \), the antiderivative found was \( -\frac{1}{2} \csc \theta + C \). By taking the derivative of this expression, we get \( \frac{1}{2} \csc \theta \cot \theta \). This result perfectly matches the original integrand \( \frac{\csc \theta \cot \theta}{2} \), validating our solution. Differentiation verification not only affirms correctness but also reinforces understanding of how differentiation and integration are inverse processes. Always incorporate this step to enhance accuracy and confidence in your integration results.