Problem 68 - 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. $$\int\left(1-\cot ^{2} x\right) d x$$

Step-by-Step Solution

Answer

The antiderivative is \( 2x + \cot x + C \).

Step 1: Rewrite the Integrand

To make integration easier, let's rewrite the integrand using a trigonometric identity. Note that \( \cot^2 x = \csc^2 x - 1 \). Thus, \( 1 - \cot^2 x = 1 - (\csc^2 x - 1) = 2 - \csc^2 x \). The integral becomes: \[ \int (2 - \csc^2 x) \, dx \].

Step 2: Integrate Term by Term

Integrate each term separately. The integral of 2 with respect to \( x \) is \( 2x \), and the integral of \( \csc^2 x \) is \( -\cot x \). Thus, the antiderivative is: \[ 2x + \cot x + C \], where \( C \) is the constant of integration.

Step 3: Verify by Differentiation

Differentiate the obtained antiderivative \( 2x + \cot x + C \) to verify it. The derivative of \( 2x \) is 2, the derivative of \( \cot x \) is \( -\csc^2 x \), and the derivative of \( C \) is 0. Therefore, the derivative is: \( 2 - \csc^2 x \), which simplifies to \( 1 - \cot^2 x \). This matches the original integrand, verifying our solution.

Key Concepts

Antiderivative

The concept of an antiderivative is fundamental in calculus. An antiderivative of a function is another function whose derivative gives the original function. In other words, if you take the derivative of the antiderivative, you get back to where you started. This is why it's also called an "indefinite integral," because it represents a whole family of functions, as opposed to a definite integral, which produces a specific numerical result. In this exercise, the antiderivative was found for the function \( 1 - \cot^2 x \). By rewriting it using a trigonometric identity, the expression was simplified to \( 2 - \csc^2 x \), which was then integrated term by term. The result, \( 2x + \cot x + C \), where \( C \) is the constant of integration, represents the most general form, covering all possible antiderivatives for the function.

Trigonometric Identity

Trigonometric identities are used to transform and simplify expressions in calculus and other areas of mathematics. In this particular exercise, the identity \( \cot^2 x = \csc^2 x - 1 \) was crucial. By using this identity, the integrand \( 1 - \cot^2 x \) was rewritten as \( 2 - \csc^2 x \). This simplified integrand made it easier to find the antiderivative. Simplifying complex trigonometric expressions in this way can often lead to easier integrations and, subsequently, clearer solutions. Recognizing and applying these identities is a powerful tool in solving calculus problems.

Integration Verification

Verification of an integration result by differentiation ensures the correctness of the solution. After computing the antiderivative, it's important to differentiate it to check whether the original function is retrieved. For the antiderivative \( 2x + \cot x + C \), its derivative was calculated and found to be \( 2 - \csc^2 x \), which simplifies back to \( 1 - \cot^2 x \). This step confirmed that the integration was performed correctly. Such verification is a critical habit to develop when learning calculus, as it ensures not only that solutions are correct but also that they deepen the understanding of the relationship between derivatives and integrals.