Problem 41 - 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. $$\int \frac{\csc \theta \cot \theta}{2} d \theta$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

We start with the given integral: \( \int \frac{\csc \theta \cot \theta}{2} d \theta \). It's often useful to recognize common derivatives when working with integrals.

Step 2: Factor Out Constant

The constant \( \frac{1}{2} \) can be factored out of the integral: \( \frac{1}{2} \int \csc \theta \cot \theta \, d \theta \).

Step 3: Recognize the Derivative

Recall the derivative of \( \csc \theta \) is \( -\csc \theta \cot \theta \). Thus, \( \int \csc \theta \cot \theta \, d \theta = -\csc \theta + C \), where \( C \) is the constant of integration.

Step 4: Apply the Constant

Now, multiply back the constant factored out earlier: \( \frac{1}{2} (-\csc \theta + C) = -\frac{1}{2}\csc \theta + \frac{C}{2} \).

Step 5: Check by Differentiation

Differentiate \( -\frac{1}{2} \csc \theta + \frac{C}{2} \) to verify. The derivative is \( \frac{1}{2} \csc \theta \cot \theta \), which matches the integrand, confirming our solution is correct.

Key Concepts

Indefinite Integrals

Indefinite integrals, also known as antiderivatives, are a foundational concept in calculus. They represent a family of functions whose derivative is the given function. Unlike definite integrals, they do not have upper and lower limits of integration. As a result, they include a constant of integration, usually denoted as "C", to account for any constant that could have been present before differentiation.

When finding an indefinite integral, we seek a function whose derivative would return us to the original function, the integrand. It helps to recognize common patterns and derivatives, for example, the derivative of \(\csc \theta\) is \(-\csc \theta \cot \theta\). This recognition simplifies the process since reversing these known derivatives can lead us directly to the antiderivative of the given function.

For our exercise, we needed to find the indefinite integral of \(\frac{1}{2} \csc \theta \cot \theta\). Recognizing the derivative of \(\csc \theta\), we can quickly determine the integral, leading to the expression \(-\frac{1}{2} \csc \theta + \frac{C}{2}\). Remember, the constant \(C\) represents any shift upwards or downwards that doesn't affect the derivative being the original integrand.

Trigonometric Functions

Trigonometric functions often appear in calculus problems related to both differentiation and integration. The main trigonometric functions include sine, cosine, tangent, cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). They describe ratios of sides in right-angled triangles and are pivotal in modeling periodic phenomena.

In calculus, we frequently encounter these functions and their derivatives or integrals, which involve understanding their relationships. For instance, understanding that the derivative of \(\csc \theta\) is \(-\csc \theta \cot \theta\) can simplify integrating functions involving these expressions.

Integrals of trigonometric functions often involve recognizing these derivative/integral pairs, which can make otherwise complex integrals straightforward. Thus, having a good grasp of these functions' properties is essential, as it simplifies the process of finding indefinite integrals and verifying those integrals through differentiation.

Differentiation Verification

Verification by differentiation is a crucial step in confirming the correct antiderivative or indefinite integral has been found. Once a candidate for the antiderivative is identified, differentiating it should yield the original integrand. This step ensures no mistakes were made in the integration process.

For example, with our exercise of integrating \(\frac{1}{2} \csc \theta \cot \theta\), we found the antiderivative to be \(-\frac{1}{2} \csc \theta + \frac{C}{2}\). By differentiating this expression, we confirmed the differentiation returned \(\frac{1}{2} \csc \theta \cot \theta\), which matches the original integrand.

This method tests the accuracy of the integration process and helps catch errors early. It reinforces understanding and intuition about the interplay between differentiation and integration, and why the constant of integration is a necessary component of indefinite integrals.