Problem 52 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int\left(1-\cot ^{2} x\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Recognize Integral Identity

The given integral is \( \int (1 - \cot^2 x) \, dx \). Recognize that \( 1 - \cot^2 x \) can be rewritten using the trigonometric identity \( \cot^2 x = \csc^2 x - 1 \). Thus, \( 1 - \cot^2 x = 1 - (\csc^2 x - 1) = 2 - \csc^2 x \).

Step 2: Break Down and Integrate

The integral \( \int (2 - \csc^2 x) \, dx \) can be split into two separate integrals: \( \int 2 \, dx - \int \csc^2 x \, dx \).

Step 3: Solve Each Integral

Calculate the integral \( \int 2 \, dx = 2x + C_1 \). For the second part, note that \( \int \csc^2 x \, dx = -\cot x + C_2 \). Combine these solutions: \( 2x - \cot x + C \) where \( C = C_1 + C_2 \) is a constant.

Step 4: Verify by Differentiation

Differentiate the result to verify. The derivative \( \frac{d}{dx}(2x - \cot x + C) = 2 + \csc^2 x \), which simplifies to the original integrand \( 1 - \cot^2 x \), confirming our solution is correct.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus as they allow us to simplify and manipulate expressions involving trigonometric functions. In calculus problems like this, recognizing these identities can simplify an otherwise complex expression. Here, the identity \( \cot^2 x + 1 = \csc^2 x \) was key. By rearranging this identity into \( \cot^2 x = \csc^2 x - 1 \), the original expression \( 1 - \cot^2 x \) in the integral becomes \( 2 - \csc^2 x \). This transformation is crucial since it turns the original problem into something more manageable. When you work with trigonometric identities, look for opportunities to turn subtractions and additions into simpler forms that are easier to integrate.

Indefinite Integrals

Indefinite integrals, often referred to as antiderivatives, represent a family of functions. They allow us to find the function that, when differentiated, will yield the original integrand. The process involves recognizing known integral forms and applying the integral rules appropriately. Here, the given integral \( \int (1 - \cot^2 x) \, dx \) was transformed into \( \int (2 - \csc^2 x) \, dx \). The problem was then split into simpler integrals: \( \int 2 \, dx \) and \( \int -\csc^2 x \, dx \). Each part is integrated separately:

• \( \int 2 \, dx = 2x + C_1 \)
• \( \int -\csc^2 x \, dx = -\cot x + C_2 \)

The sum of these results yields the general antiderivative: \( 2x - \cot x + C \), where \( C \) represents an arbitrary constant, combining \( C_1 \) and \( C_2 \). Every antiderivative solution can include such constants, signifying the indefinite nature of these integrals.

Verification by Differentiation

Verification by differentiation is a crucial step to ensure the correctness of a calculated antiderivative. After integrating, differentiate your result to check if you can retrieve the original integrand. This not only validates your answer but also reinforces your understanding of the interplay between differentiation and integration. In this exercise, taking the derivative of the antiderivative form \( 2x - \cot x + C \) involves:

• Differentiating \( 2x \) yields \( 2 \).
• Differentiating \(-\cot x\) involves applying the derivative formula of cotangent: \( \csc^2 x \).
• The constant \( C \) vanishes after differentiation since the derivative of a constant is zero.

Thus, the expression emerges as \( 2 - \csc^2 x \), verifying it simplifies back to the original expression \( 1 - \cot^2 x \). This consistent loop verifies our solution's accuracy, providing confidence in the calculated indefinite integral.