Problem 40 - Calculus Single Variable

Question

Use trigonometric identities to compute the indefinite integrals. $$ \int \cot ^{2}(x) d x $$

Step-by-Step Solution

Answer

The integral is \( \int \cot^2(x) \, dx = -\cot(x) + x + C \).

Step 1: Recall Trigonometric Identity

Start by recalling the trigonometric identity for cotangent squared. We know that: \[ \cot^2(x) = \csc^2(x) - 1 \] This is derived from the identity \[ \csc^2(x) = 1 + \cot^2(x) \].

Step 2: Substitution in Integral

Substitute \( \cot^2(x) \) with \( \csc^2(x) - 1 \) in the integral:\[ \int \cot^2(x) \, dx = \int (\csc^2(x) - 1) \, dx \] This breaks down to two separate integrals: \[ \int \csc^2(x) \, dx - \int 1 \, dx \].

Step 3: Integrate Each Term

Integrate each term separately. The integral of \( \csc^2(x) \) is a known result:\[ \int \csc^2(x) \, dx = -\cot(x) + C_1 \]where \( C_1 \) is a constant of integration.The integral of 1 with respect to \( x \) is:\[ \int 1 \, dx = x + C_2 \]where \( C_2 \) is another constant of integration.

Step 4: Combine the Results

Combine the results of the integrals:\[ \int \cot^2(x) \, dx = (-\cot(x) + x) + (C_1 - C_2) \]Combine constants: let \( C = C_1 - C_2 \), then the final answer is:\[ \int \cot^2(x) \, dx = -\cot(x) + x + C \]

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools for simplifying and solving integrals, especially when dealing with trigonometric functions. In this case, we focus on an identity involving the cotangent function.One important identity to remember is: \[ \csc^2(x) = 1 + \cot^2(x) \]This can be rearranged to express \( \cot^2(x) \) as:\[ \cot^2(x) = \csc^2(x) - 1 \]This transformation allows the original integral to be rewritten in a more manageable form. Understanding these basic relationships between functions is critical in both trigonometry and calculus. With these identities, we can take a complex problem and break it down into simpler parts, making it easier to solve.

Integration Techniques

Integration techniques are key in evaluating integrals, especially when direct integration isn't straightforward. When solving the integral of \( \cot^2(x) \), we utilized an important technique: substitution using trigonometric identities.Here’s how the process works:

• First, identify an equivalent expression for the function, such as \( \csc^2(x) - 1 \), using trigonometric identities.
• Then, break down the integral into simpler terms: \( \int \csc^2(x) \, dx \) and \( \int 1 \, dx \).

This method offers a strategic approach by dividing the problem, making each part easier to handle. After substitution, these simpler integrals can be computed using known results, thereby solving the initial problem efficiently.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), is the reciprocal of the tangent function. It plays a vital role in calculus, particularly in integration problems involving trigonometric expressions.In the given exercise, the integral of the square of the cotangent function, \( \cot^2(x) \), is computed. By breaking it down through a trigonometric identity, we avoid directly integrating a more complex expression. Instead, we handle parts separately and utilize known integrals:

• The integral of \( \csc^2(x) \) is a standard result: \( \int \csc^2(x) \, dx = -\cot(x) + C \).
• The integral of 1 is simply: \( \int 1 \, dx = x + C \).

Understanding the role of \( \cot(x) \) and its relationship with other trigonometric functions is crucial not only to solve the integral efficiently but also to grasp broader concepts in trigonometry and calculus.