Problem 53 - Thomas Calculus

Question

In Exercises \(17-56,\) 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. $$\begin{array}{l}{\int \cot ^{2} x d x} \\ {\text {(Hint: } 1+\cot ^{2} x=\csc ^{2} x )}\end{array}$$

Step-by-Step Solution

Answer

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

Step 1: Use the Trigonometric Identity

Given the identity \(1 + \cot^2 x = \csc^2 x\), we can substitute \(\cot^2 x\) with \(\csc^2 x - 1\) in the integral, which gives us: \[ \int \cot^2 x \ dx = \int (\csc^2 x - 1) \ dx. \]

Step 2: Split the Integral

Rewrite the integral using the linearity of integration to separate the terms: \[ \int (\csc^2 x - 1) \ dx = \int \csc^2 x \ dx - \int 1 \ dx. \]

Step 3: Integrate Each Term Separately

Integrate \(\csc^2 x\) and \(1\) separately. The antiderivative of \(\csc^2 x\) is \(-\cot x\) and the antiderivative of \(1\) is \(x\). So we have: \[ \int \csc^2 x \ dx = -\cot x + C_1 \] and \[ \int 1 \ dx = x + C_2, \] where \(C_1\) and \(C_2\) are constants of integration.

Step 4: Combine the Results

Combine the results of these integrals: \[ \int \cot^2 x \ dx = -\cot x - x + C, \] where \(C = C_1 + C_2\) is the general constant of integration.

Step 5: Check by Differentiation

Differentiate the result, \(-\cot x - x + C\), to verify the antiderivative. The derivative of \(-\cot x\) is \(\csc^2 x\), and the derivative of \(-x\) is \(-1\). Thus, we have: \(\csc^2 x - 1 = \cot^2 x\), which matches the original integrand.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables for which both sides of the identity are defined. These identities are incredibly useful in calculus, as they allow us to simplify complex expressions and solve integrals that may not initially appear manageable.
One of the most commonly used identities in this exercise is

• \(1 + \cot^2 x = \csc^2 x\)

This identity relates the cotangent and cosecant functions. It helps transform the integral of \(\cot^2 x\) into something more feasible to integrate by substituting \(\cot^2 x\) with \(\csc^2 x - 1\). This simplifies the process and forms the basis for determining the antiderivative.

Antiderivative

An antiderivative is a function whose derivative is the given function. In other words, the process of finding an antiderivative is the reverse of differentiation. For indefinite integrals, finding the antiderivative means discovering what original function had the given expression as its derivative.
In this particular exercise, we are looking for the antiderivative of \(\cot^2 x\). By using our trigonometric identity, we rewrite it as \(\csc^2 x - 1\), which we can split into parts that have known antiderivatives:

• The antiderivative of \(\csc^2 x\) is \(-\cot x\).
• The antiderivative of \(1\) is \(x\).

This results in the general antiderivative for the integral, \(-\cot x - x + C\), where \(C\) is a constant of integration.

Differentiation

Differentiation is the process of finding the derivative of a function, which shows how the function changes as its input changes. It’s the reverse operation of integration. To verify our calculated antiderivative, we differentiate our resulting function.
For \(-\cot x - x + C\), we check by deriving:

• The derivative of \(-\cot x\) is \(\csc^2 x\).
• The derivative of \(-x\) is \(-1\).

Combining these results, we recover the original function’s derivative, \(\csc^2 x - 1\), which simplifies back to \(\cot^2 x\). This confirms the correctness of our antiderivative calculation.

Linearity of Integration

Linearity of integration, also known as the superposition principle, states that the integral of a sum of functions is the sum of the integrals of each function. This property is incredibly useful when breaking down complex integrals into simpler parts that are easier to solve individually.
In our exercise, we utilize this principle to separate

• \(\int (\csc^2 x - 1) \ dx\)

into

• \(\int \csc^2 x \ dx - \int 1 \ dx\)

This separation allows each term to be integrated on its own using known antiderivatives, simplifying the process greatly and making the solution accessible even in more complex integrals.